Determinations  of  Stellar  Parall 


BV 


HENRY  NORRIS  RUSSELL 

Professor  of  Astronomy  at  Princeton  U trivet 


BASED  UPON  PHOTOGRAPHS  TAKEN  AT  THE  CAMBRIDGE  OBSERVATORY 
nv  ARTHUR  R.  HINKS  AND  THE  WRITER 


WITH  MAGNITUDES  AND  SPECTRA  DETERMINED  AT  THE  HARVARD  COLLEGE 
OBSERVATORY  UNDER  DIR  i  >      •  k    0 


PUBLISHED  BV  THE  C 


HINGTON,  D.  C. 

ND;!!;  INSTITUTION  OF  WASHINGTON 

1911 


Determinations  of  Stellar  Parallax 


BY 


HENRY  NORRIS  RUSSELL 

Professor  of  Astronomy  at  Princeton  University 


BASED  UPON  PHOTOGRAPHS  TAKEN  AT  THE  CAMBRIDGE  OBSERVATORY 
BY  ARTHUR  R.  HINKS  AND  THE  WRITER 


WITH  MAGNITUDES  AND  SPECTRA  DETERMINED  AT  THE  HARVARD  COLLEGE 
OBSERVATORY  UNDER  DIRECTION  OF  PROFESSOR  E.  C.  PICKERING 


WASHINGTON,  D.  C. 

PUBLISHED  BY  THE  CARNEGIE  INSTITUTION  OF  WASHINGTON 

1911 


X/3 

Astron.  Oept. 


CARNEGIE  INSTITUTION  OF  WASHINGTON 
PUBLICATION  No.  147 


ASTRONOMY 


PRESS  OF  GIBSON  BROTHER* 
WASHINGTON,  D.  C. 


CONTENTS. 

PAOB. 

INTRODUCTION: 

History  of  the  work.    Acknowledgements i 

CHAPTER  I.  Methods  of  Observation  and  Measurement: 

1.  General  policy 3 

2.  Reasons  for  taking  separate  plates 3-4 

3.  Hour-angle  error  and  its  elimination 4-6 

4.  Observation  of  bright  stars.     Color-screen 6-9 

5.  Other  systematic  and  quasi-systematic  errors 9 

6.  The  photographic  observations 10-12 

7.  Measurement  of  the  plates 12-13 

8.  Errors  of  the  measuring  apparatus 13-15 

9.  Economy  of  measurement 15-16 

10.  Working  list 16-17 

1 1 .  Magnitudes  and  spectra 17 

CHAPTER  II.  Reduction  of  the  Measures: 

1 .  Formula?  and  standard 1 9-20 

2.  Dyson's  method  for  finding  the  plate-constants 21 

3.  Accuracy  of  this  method 2 1-25 

4.  Number  of  comparison-stars 25-27 

5.  Solution  for  the  parallax  and  proper-motion 27-28 

CHAPTER  III.  The  Observations: 

1.  General  summary 29 

2-6.  Description  of  Table  C 

2.  Stars  observed,  and  standard 29- 30 

3.  Data  for  individual  plates 30-32 

4.  Approximate  solution  for  all  stars 33 

5.  I<east-squares  solution  for  the  principal  stars 33~35 

6.  Stars  observed  at  but  two  epochs 35~36 

7.  Success  in  eliminating  hour-angle  error 36-38 

8.  Examples  of  details  of  measurement  and  reduction 38-40 

9.  Example  of  approximate  solution 41 

10.  Case  when  a  comparison-star  has  sensible  proper-motion 4i~43 

CHAPTER  IV.  Discussion  of  the  Observations: 
I.  Absence  of  Systematic  Errors. 

1 .  Errors  of  observation  almost  wholly  accidental  in  character 45 

2.  Search  for  systematic  error  depending  on  position  on  the  plate 45~46 

3.  Search  for  error  depending  on  magnitude 46~47 

4.  Search  for  error  depending  on  spectral  type 47~4^ 

5.  Search  for  error  depending  on  right  ascension 48~49 

6.  Conclusion :  Systematic  errors  apparently  insensible 49-5° 

II.  The  Proper-Motions: 

7.  Comparison-stars  with  sensible  proper-motion 5° 

8.  Average  proper-motion  of  the  rest 5°~53 

9.  Proper-motion  of  the  parallax-stars;  reality  of  the  observed  corrections .  53-55 
III.  The  "Two- Epoch"  Parallaxes: 

Their  probable  errors 55~56 


10.  Reliability  of  these  resultss  .  Their  pro 

•  ,  V  f  '"  --'"  1  J>          ^W 


111 


iT  CONTENTS. 

CHAPTER  IV.  Discussion  of  the  Observations — Continued. 

IV.  Accidental  Errors  of  Star-Places: 

1 1 .  Types  of  error.     Notation 57~58 

12.  Error  of  measurement 58 

13.  Error  peculiar  to  the  individual  image 59~6° 

14.  Error  peculiar  to  the  individual  plate 60-61 

15.  Error  due  to  instrumental  or  seasonal  changes 61-63 

1 6.  Summary  of  results.     Best  number  of  exposures  per  plate 63-65 

V.  Probable  Errors  for  the  Parallax-Stars: 

1 7.  Loss  of  accuracy  for  close  double  stars 65 

1 8.  Dependence  of  accuracy  upon  photographic  magnitude,  and  position 

on  the  plate 66-68 

19.  Importance  of  correct  exposure 69 

20.  Results  for  stars  close  together  only  partially  independent.     Explana- 

tion    70-71 

21.  Comparison  of  the  average  precision  attained  by  different  observers .  .  .  .    71-72 

CHAPTER  V.  Results  of  Observation: 

I.  Results  of  the  Present  Work: 

1.  Description  of  Table  A 73~74 

TABI.E  A.     Observed  Parallaxes 76~77 

Notes  to  Table  A 74.  75.  77 

2.  Reality  of  the  results.     Negative  parallaxes 77~78 

II.  Comparison  with  other  Observers: 

3.  Description  of  Table  B 79~8o 

TABLK  B.     Results  of  other  observers 80-81 

Notes  to  Table  B 81-82 

4.  Search  for  systematic  error.     Average  value  of  systematic  errors  for 

most  observers  very  small 83-88 

III.  Comparison  with  Kapteyn's  Formulae: 

5.  Parallax  of  the  comparison-stars 88-89 

6.  Data  for  the  individual  parallax-stars 89 

7.  Comparison  by  groups.  Systematic  differences  for  different  spectral  types  90-92 

IV.  Astrophysical  Data: 

8.  Brightness  and  cross-velocity  of  the  individual  stars 92-94 

9.  Means  for  different  spectral  types 94~96 

10.  Possible  explanations  of  the  differences 96-97 

1 1.  Bearing  on  stellar  evolution 97~9§ 

12.  Masses  of  binary  stars 98-101 

13.  Distribution  of  stars  of  different  spectral  types 101-103 

TABUS  C.     Details  of  the  Observations 104-142 


Determinations  of  Stellar  Parallax 


INTRODUCTION. 

The  observations  upon  which  the  present  volume  is  based  were  made  at 
the  Cambridge  Observatory  (England)  in  the  years  1903-1907,  according 
to  plans  prepared  for  the  stellar  parallax  work  of  that  observatory  by  Mr. 
Arthur  R.  Hinks,  chief  assistant  at  the  observatory,  and  the  writer,  who  had 
been  appointed  a  research  assistant  of  the  Carnegie  Institution  of  Washington 
for  the  prosecution  of  the  work  whose  results  are  here  given. 

It  was  originally  planned  that  all  the  observations  should  be  made  by 
the  writer,  but  in  the  autumn  of  1904  he  was  disabled  by  serious  illness,  and 
Mr.  Hinks  very  kindly  undertook  their  continuance.  Upon  the  writer's 
recovery,  his  term  as  a  research  assistant  had  practically  expired  and  he 
received  a  call  to  America  for  the  following  year.  It  was  then  arranged  that 
Mr.  Hinks  should  complete  the  photographic  work  (for  which  he  and  the 
writer  are  therefore  jointly  responsible  in  nearly  equal  proportions — 43  per 
cent  being  done  by  the  former  and  57  per  cent  by  the  latter). 

The  methods  of  observation  and  measurement,  detailed  in  Chapter  I— 
so  far  as  they  contain  anything  new — are  also  the  result  of  collaboration 
between  Mr.  Hinks  and  the  writer. 

For  the  measurement  and  reduction  of  the  plates,  the  discussion  of  the 
results,  and  the  conclusions  deduced  therefrom  (which  together  occupy  the 
remaining  chapters  of  this  volume)  the  writer  is  alone  responsible. 

The  magnitudes  and  spectra  of  the  stars  have  been  determined  at  the 
Harvard  College  Observatory,  under  the  direction  of  Prof.  E.  C.  Pickering, 
to  whom  the  writer  is  very  greatly  indebted  for  this  valuable  and  generous 
contribution. 

It  is  also  a  pleasure  to  him  to  express  his  gratitude  to  Sir  Robert  Ball, 
Director  of  the  Cambridge  Observatory,  and  to  the  Observatory  Syndicate, 
who  did  everything  in  their  power  to  facilitate  his  work;  to  the  members  of 
the  Observatory  staff,  for  their  cordial  interest,  and  in  particular  to  Mr. 
Hinks,  for  valuable  comment  and  criticism  and  many  kindnesses  in  addition 
to  the  invaluable  collaboration  already  described ;  and  finally  to  the  authori- 
ties of  Princeton  University,  for  the  time  and  instrumental  means  for  com- 
pleting the  work  there. 


CHAPTER  I. 
METHODS  OF  OBSERVATION  AND  MEASUREMENT. 

§  i .  General  Policy* 

The  object  of  the  present  work  is  the  determination  of  the  parallax  of  cer- 
tain selected  stars  relative  to  the  mean  of  a  group  of  comparison-stars  in  their 
neighborhood,  and  incidentally  of  the  several  comparison-stars  relative  to  their 
mean.  In  preparing  plans  for  it  two  main  objects  were  constantly  in  view : 

a.  To  eliminate  at  any  cost  all  known  sources  of  systematic  error. 

b.  To  secure  the  most  advantageous  relation,  consistent  with  the  first 
condition,  between  the  amount  of  labor  to  be  expended  and  the  probable 
accuracy  of  the  results  to  be  obtained. 

§2.  Reasons  for  taking  Separate  Plates. 

After  careful  consideration,  it  was  decided  not  to  adopt  Professor  Kap- 
teyn's  plan  of  making  exposures  upon  the  same  plate  at  three  successive 
epochs  of  maximum  parallactic  displacement  separated  by  approximately  six 
months  each.  The  reasons  for  this  decision  were  as  follows : 

(1)  The  identification  and  preservation  of  the  undeveloped  plates  and 
their  readjustment  upon  the  telescope  in  exactly  the  right  position  (so  that 
the  later  series  of  star-images  shall  neither  interfere  with  the  earlier  ones 
nor  be  too  far  from  them)  seriously  increase  the  labor  of  observation. 

(2)  Failure  to  obtain  satisfactory  results  from  the  exposures  of  any  one 
epoch  renders  the  whole  plate  useless,  at  least  for  discussion  according  to 
Kapteyn's  method. 

(3)  It  is  necessary  with  this  method  to  wait  at  least  a  year  after  beginning 
observations  before  any  plates  can  be  measured.     (As  the  writer's  appoint- 
ment as  a  research  assistant  of  the  Carnegie  Institution  was  for  two  years 
only,  this  was  in  itself  a  conclusive  argument.) 

That  these  anticipations  were  in  accordance  with  the  facts  appears  from 
the  recently  published  work  of  Kostinsky,  who  finds  also,  in  results  obtained 
by  this  method,  clear  evidence  of  a  systematic  error  (affecting  the  deduced 
parallaxes  and  varying  with  the  magnitude  of  the  stars)  between  the  results 
from  plates  on  which  the  first  exposure  was  made  at  a  maximum  and  at  a 
minimum  of  parallactic  displacement.!  The  source  of  this  error  appears  to 
be  the  alteration  of  the  latent  image  of  a  star  during  the  long  interval  between 
the  earlier  exposure  and  the  time  of  development!  and  also  the  influence  of 
neighboring  images,  at  least  for  the  bright  stars.§ 

'Compare  the  discussion  by  Hinks  and  Russell  in  Monthly  Notices,  LXV,  pp.  775-787,  upon  which  much 
of  this  chapter  is  based  and  which  is  often  quoted  verbatim. 

fPubl.  de  1'  Obs.  Cent.  Nicolas,  S&ie  n,  vol.  17,  part  2  (1905),  p.  138.         JDitto,  p.  69. 
§Tikhoff;  Pulkovo  Mittheilungen,  No.  18,  p.  101. 


4  DETERMINATIONS  OF   STELLAR  PARALLAX. 

The  experience  of  these  observers  therefore  justifies  the  decision,  reached 
before  Uie  publication  of  their  results,  with  respect  to  the  present  work, 
namely : 

RULE  I :  Take  separate  plates  at  each  epoch,  and  develop  them  at 
once. 

In  this  case  it  is  usually  possible,  when  a  plate  turns  out  to  be  bad,  to 
duplicate  it  within  a  few  days ;  and,  under  favorable  conditions,  several  plates 
can  be  obtained  at  one  epoch,  even  under  the  severe  restrictions  demanded  by 
parallax  work. 

If  no  observations  at  all  can  be  secured  before  the  star  becomes  unobserv- 
able,  those  of  a  year  later  will  fill  the  gap,  with  no  inconvenience  beyond  the 
delay.  It  is  in  fact  desirable  to  have  some  stars  under  observation  at  every 
available  season  for  a  considerable  time,  as  in  this  way  the  uniformity  of  the 
instrumental  conditions  throughout  the  period  can  be  controlled. 

§3.  Hour-angle  Error  and  its  Elimination. 

Of  the  various  systematic  errors  which  may  affect  the  positions  of  photo- 
graphic star-images,  the  most  serious,  from  our  standpoint,  are  those  included 
by  Professor  Kapteyn*  in  the  category  of  "hour-angle  error."  Under  this 
head  come  all  errors  which  tend  to  alter  the  relative  coordinates  of  the  images, 
when  the  same  stars  are  observed  at  different  hour-angles.  One  cause  of 
this  may  be  optical  distortion  varying  with  the  hour-angle,  which  is  espe- 
cially to  be  feared  when,  as  at  Cambridge,  a  mirror  forms  part  of  the  optical 
train.  This  may  displace  bright  stars  relatively  to  faint  ones,  or  stars 
in  one  part  of  the  plate  relatively  to  those  in  another.  Still  more  serious, 
because  certainly  unavoidable,  is  the  influence  of  atmospheric  dispersion. 
The  refraction-constant  for  different  stars  varies  with  the  mean  photo- 
graphically effective  wave-length  of  their  light  and,  as  Kapteyn  has  suggested 
and  Bergstrandj  has  recently  shown  by  experiment,  the  latter  varies  not 
only  with  the  spectral  type  of  the  stars,  but  with  their  brightness  and  even 
with  the  length  of  exposure  for  the  same  star.  Hence  there  arise  relative 
displacements  of  the  star-images  varying  with  the  zenith-distance,  and  con- 
sequently with  the  hour-angle. 

It  is  therefore  of  prime  importance  (as  was  first  pointed  out  by  KapteynJ) 
that  in  investigations  of  stellar  parallax  all  the  photographs  of  a  given  region 
shall  be  taken  at  the  same  hour-angle,  or  at  least  that  the  mean  hour-angles 
for  the  epochs  of  positive  and  negative  parallactic  displacement  shall  be 
the  same. 

To  observe  a  given  star  only  at  a  fixed  hour-angle  means,  however, 
that  the  morning  observations  must  be  begun  later  or  the  evening  ones 
finished  earlier  than  the  most  favorable  dates  (and  in  most  cases  both). 
This  involves  a  considerable  loss  of  parallax  factor  and  consequently  of 

•Cfoninfen  Pub.  No.  i.  p.  67.  Jlbid..  p.  68.  fAitr.  Ntchrichten  3999  and  4240. 


METHODS  OF   OBSERVATION  AND  MEASUREMENT. 


the  weight  of  the  determination  of  parallax ;  but  this  is  a  much  less  serious 
matter  than  the  introduction  of  systematic  error. 

The  question  next  arises.  Should  this  hour-angle  be  different  for  differ- 
ent stars,  or  the  same  for  all  (in  which  case  they  should  obviously  be  observed 
on  the  meridian)  ? 

The  parallax  factor  in  x,  that  is,  in  right  ascension,  for  any  star,  is 
independent  of  its  declination  and  depends  only  on  its  right  ascension  and  the 
sun's  longitude.  Its  maximum  value  varies  slightly  with  the  right  ascension 
of  the  star,  from  i.oo  in  6h  and  i8h  to  0.92  in  oh  and  I2h.  But  when  the  par- 
allax factor  in  #  is  a  maximum,  the  star  is  in  quadrature  with  the  sun,  and 
it  is  usually  impossible  to  observe  it  on  the  meridian.  The  available  paral- 
lactic  displacement  for  meridian  observations  is  therefore  diminished  by  an 
amount  which  varies  with  the  season  of  the  year  at  which  the  observations 
are  made,  and  depends  upon  the  star's  right  ascension  and  on  the  latitude 
of  the  observer. 

Table  i*  is  computed  for  Cambridge  and  represents,  on  account  of  the 
high  latitude,  a  somewhat  extreme  case.  It  givesfor  different  right  ascensions : 

(1)  The  dates  between  which  a  star  can  be  observed  on  the  meridian, 
while  the  sun  is  more  than  10°  below  the  horizon. 

(2)  The  parallax  factors  in  x  for  these  limiting  dates. 

(3)  The  total  available  parallactic  displacement  in  x,  in  terms  of  the 
star's  heliocentric  parallax. 

(4)  The  dates  and  amounts  of  the  maximum  parallactic  displacements 
in  x  in  each  direction,  in  terms  of  the  same  unit. 

TABLE  i. — Observations  on  the  Meridian. 


Limiting  dates. 

Parallax  factor. 

Available 

Maximum  displacement. 

R.A. 

Morning  : 
after. 

Evening: 
before. 

Morning. 

Evening. 

displace- 
ment. 

Morning. 

Evening. 

Amount. 

o" 

Aug.    2 

Jan.     3 

+0.65 

-0.89* 

•57 

June  22 

Dec.  22 

0.92 

2 

4 

Aug.  2) 
Sept.  14 

Jan.  25 
Feb.  16 

0.89 

o  94* 
0.98 

7" 

.87 

July  22 
Aug.  22 

Jan.  20 
Feb.  18 

o  94 
0.98 

6 

Oct.    4 

Mar.   9 

0.96 

0.97 

93 

Sept.  23 

Mar.  2  1 

i  .00 

8 

Oct.  24 

Mar.  30 

o  97 

0.90 

.87 

Oct.  25 

Apr.  21 

0.98 

10 

Nov.  15 

Apr.  20 

o  94* 

o  79 

•73 

Nov.  24 

May  22 

o  94 

12 

Dec.    9 

May    9 

0.88* 

0.69 

.61 

Dec.  22 

June  22 

0.92 

>4 

Jan.     5 

May  29 

0.88* 

o  57 

•51 

Jan.  20 

July  22 

°  94 

16 

Feb.  17 

June  2  1 

0.98* 

°  47 

45 

Feb.  18 

Aug.  22 

0.98 

17 

Mar.  20 

July    9 

°  95 

0.52 

•47 

Mar.    5 

Sept.  7 

i  .00 

18 

Apr.  30 

Aug.    4 

°  77 

0.71 

.48 

Mar.  21 

Sept.  23 

1.00 

>9 

June    3 

Sept.  1  6 

0.54 

°  94 

•48 

Apr.    6 

Oct.    9 

i  .00 

20 

June  20 

Oct.  25 

0.48 

0.98 

.46 

Apr.  21 

Oct.  25 

0.98 

22 

July  14 

Dec.    6 

o  55 

0.90* 

49 

May  22 

Nov.  24 

o  94 

Sometimes  one  of  these  maxima  falls  between  the  limiting  dates.  In 
such  cases  the  parallax  factor  at  one  of  the  limiting  dates  is  not  the  greatest 
available,  and  is  marked  with  an  asterisk  in  the  table. 

•Compare  Hinks,  Monthly  Notices,  LVIII,  p.  440;  and  the  similar  tables  by  De  Sitter  for  different  lati- 
tudes in  Groningen  Pub.  No.  15,  pp.  5-10;  which,  however,  are  constructed  for  the  Sun  12°  below  the  horizon. 


DETERMINATIONS  OF   STELLAR   PARALLAX. 


It  will  be  noticed  that  the  parallax  factors  for  the  morning  and  evening 
observations  are  usually  unequal.  By  making  the  observations  off  the  merid- 
ian, in  an  hour-angle  constant  for  each  star,  but  different  for  stars  of  differ- 
ent right  ascension,  symmetrical  parallax  factors  for  the  morning  and  evening 
observations  can  be  obtained.* 

Table  2  gives  the  hour-angle  for  such  symmetrical  observations  and  the 
available  parallactic  displacement  in  terms  of  the  heliocentric  parallax : 

TABLE  2. — Observations  off  the  Meridian. 


Available 

R.  A. 
of  star. 

Hour-angle. 

Sidereal 
time  of  ob- 
servation. 

Available 
displace- 
ment. 

displace- 
ment for  ob- 
servation on 

meridian. 

o" 

-3"  33- 

3lh   38" 

•72 

•57 

2 

-'     44 

o     16 

.80 

•7" 

I 

-o    42 

0        0 

3     .8 
6      o 

M 

.87 

8 

+o    43 

8    42 

.89 

•87 

10 

+  i     44 

n     44 

.80 

•73 

13 

+3      33 

4    22 

•72 

.61 

14 

+»      5 

,6      5 

.65 

•51 

16 
18 

+  1       10 
0        0 

17      10 

18      o 

55 

47 

is 

20 

—  1       IO 

18    50 

•55 

.46 

22 

-a      5 

19    55 

.65 

49 

It  appears  upon  comparison  with  the  preceding  table  that  there  is 
on  the  average  very  little  gained  as  regards  available  parallactic  displacement 
by  going  off  the  meridian.  On  the  other  hand  the  stars  in  the  neighbor- 
hood of  i8h  right  ascension  are  inconveniently  crowded  together  as  regards 
the  sidereal  time  of  observation,  and  hence  also  as  regards  the  time  of  the 
year  which  they  must  be  observed — which  depends  upon  this  in  just  the 
same  way  as  in  the  case  of  meridian  observations.  It  was  therefore  decided 
to  follow  the  simple  rule: 

RULE  II :  All  photographs  are  to  be  taken  as  nearly  as  possible  on 
the  meridian. 

§4.  Observation  of  Bright  Stars.     Color  Screen. 

In  almost  all  forms  of  precise  astronomical  observation  systematic  errors 
are  to  be  feared  when  it  is  necessary  to  determine  the  relative  position  of 
stars  of  very  different  brightness.  In  observing  with  the  meridian  circle 
or  heliometer  this  source  of  error  is  removed  by  the  use  of  screens  placed 
before  the  object-glass  (or  one-half  of  it),  which  reduce  the  apparent  bright- 
ness of  the  brighter  star  to  approximate  equality  with  the  fainter  ones. 

In  photographic  work  a  screen,  if  used  at  all,  must  clearly  be  placed 
near  the  plate,  so  as  to  affect  the  light  of  the  bright  star  alone,  while  allowing 
unobstructed  passage  to  that  of  the  faint  stars.  At  the  Greenwich  Obser- 
vatoryt  this  has  been  accomplished  by  the  use  of  an  occulting  shutter, 


•Kapteyn,  Groningen  Pub.  I,  p.  70. 


tMonthly  Notices/ LIX,  p.  501. 


METHODS  OF   OBSERVATION  AND  MEASUREMENT.  7 

immediately  in  front  of  the  plate,  which  is  lifted  to  give  a  series  of  short 
exposures,  of  small  aggregate  duration,  upon  the  bright  star,  during  the 
progress  of  the  long  exposure  upon  the  faint  stars;  and  somewhat  similar 
devices  have  been  used  by  Schlesinger  at  the  Yerkes  Observatory.* 

The  screen  designed  by  the  writer  for  use  in  the  present  work  is  of  a 
different  character:  In  the  plate-holder,  directly  in  front  of  the  sensitive 
plate,  is  inserted  a  screen  of  clear  glass,  upon  which,  in  the  optical  center 
of  the  field,  is  a  small  patch  of  gelatin  stained  with  a  yellow  dye.  The  image 
of  the  bright  star  is  made  to  fall  upon  this  patch,  and  suffers  a  large  reduction 
in  photographic  brightness — depending  upon  the  absorption  of  the  dye  used— 
which  may  amount  to  several  magnitudes. 

An  experimental  screen,  with  a  gelatin  film  stained  as  above,  was  found 
to  cut  down  the  photographic  brightness  of  a  star  by  rather  more  than  six 
magnitudes,  while  the  sharpness  of  the  images  was  satisfactory,  although  the 
screen  was  nothing  more  than  a  selected  piece  of  plate  glass. 

It  is  well  known  that  the  principal  effect  of  interposing  a  plane-parallel 
transparent  plate,  of  thickness  t  and  refractive  index  p,  in  the  path  of  a  pencil 
of  converging  rays,  is  to  set  back  their  focus  along  the  normal  to  the  plate 

through  the  distance 1.     A  plane-parallel  glass  screen  therefore  sets  back 

the  focal  plane  of  the  telescope  by  about  one-third  of  its  own  thickness  (the 
refractive  index  of  glass  being  about  1.5).  There  is  in  addition  a  very  small 
distortion  proportional  to  the  thickness  of  the  plate  and  to  the  cube  of  the 
angle  of  incidence.  For  a  plate  of  refractive  index  i  .5  and  thickness  of  5  mm. 
the  amount  of  the  distortion  is  less  than  one  ten-thousandth  mm.  for  angles  of 
incidence  less  than  2.5°.  It  may  therefore  be  safely  neglected  in  practice. 

The  center  of  projection  of  the  field  is  shifted  by  the  action  of  the  screen 
by  exactly  the  same  amount  as  the  focal  plane,  so  that  the  interposition  of 
the  screen  has  no  effect  on  the  scale- value. 

The  effect  of  irregularities  of  the  surfaces  of  the  screen  may  easily  be 
calculated.  We  may  safely  disregard  the  variations  in  the  thickness  of  the 
screen  and  regard  the  angle  between  the  normal  to  the  surface  at  any  point 
and  the  normal  to  the  surf  ace  of  a  true  plane-parallel  plate  as  a  small  quantity, 
whose  square  may  be  neglected. 

To  this  degree  of  approximation,  the  position  of  the  focal  plane  is  un- 
altered by  the  irregularities  of  the  surface  and  the  effects  of  irregularities  of 
the  two  surfaces  are  independent  of  one  another.  If  the  thickness  of  the 
plate  is  /,  its  refractive  index  n,  and  the  distance  of  its  inner  (nearest)  surface 
from  the  focal  plane  is  s,  a  deflection  of  the  normal  to  the  outer  surface  by 
an  angle  0,  produces  a  displacement  of  the  image  in  the  focal  plane  of  mag- 
nitude 0,  GU—  i)  (s-\ — )  in  the  direction  in  which  the  inward  normal  to  the 

•Schlesinger,  Science,  N.  S.,  vol.  xxv,  p.  568. 


8  DETERMINATIONS  OF  STELLAR  PARALLAX. 

plate  is  displaced,  while  a  deflection  of  the  normal  to  the  inner  surface  by  an 
angle  6,  displaces  the  image  by  the  amount  8,  (n~i)s  in  the  direction  in 
which  the  outward  normal  to  the  plate  at  this  surface  is  displaced.  If  we 

assume 

M=i-5  1=1.5  mm.  5  =  0.5  mm. 

(which  closely  represent  the  conditions  for  the  screen  actually  used)  we 
find  that  the  displacement  of  the  image  due  to  the  irregularities  of  the  outer 

surface  will  be  less  than  mm.  if  e'<>  that  is'  if  »,<  30"  approx- 


imately, while  for  the  second  surface  we  must  have  02  <-  ;-or^  <I/3° 


approximately.     Thus  it  appears  that  the  surface  of  the  screen  need  be  by 
no  means  optically  perfect  in  order  to  avoid  sensible  distortion. 

As  the  absorbing  film  can  be  made  exceedingly  thin,  its  presence  will 
not  influence  these  results.  It  should,  however,  be  put  on  the  side  of  the 
screen  which  is  nearest  the  sensitive  plate. 

The  method  above  described  appears  to  be  more  convenient  than  the 
use  of  an  occulting  shutter  to  obscure  the  image  of  the  bright  star,  and 
also  than  the  method  of  dyeing  the  film  of  the  sensitive  plate  itself  (which 
may  introduce  distortion  of  the  film).  It  is  very  easy  to  set  the  color- 
screen  in  exactly  the  right  place  by  removing  the  back  of  the  plate-holder 
and  looking  at  the  images  of  the  stars  with  a  low-power  eye-piece.  In 
this  way  it  is  possible  to  equalize  the  images  of  a  bright  star  and  a  faint 
companion  30"  distant. 

The  chief  disadvantage  of  this  method  —  which  it  shares  with  all  others 
depending  upon  the  use  of  colored  absorbing  media  —  is  that  the  effective 
wave-length  of  the  light  of  the  bright-star  and  the  comparison-stars  is  differ- 
ent, and  hence  atmospheric  dispersion  comes  into  play  with  unusual  force. 
But  this  is  completely  eliminated  along  with  the  hour-angle  errors  by  taking 
all  photographs  upon  the  meridian.  The  rule  was  therefore  adopted: 

RULE  III:  All  stars  brighter  than  the  fifth  magnitude  are  to  be 
photographed  with  a  color  screen. 

The  screen  which  was  used  in  the  present  work  was  made  of  worked 
glass  by  Messrs.  Sanger,  Shepherd  &  Co.,  of  London.  A  thin  film  of  gelatin 
was  coated  on  the  surface  and  dyed  a  deep  orange  ;  it  was  then  cut  down  to 
a  patch  about  3  mm.  square.  This  screen  diminished  the  light  of  a  star 
about  six  magnitudes.  It  gave  very  satisfactory  definition  (the  images 
photographed  through  it  being  quite  as  sharp  as  those  of  faint  stars  obtained 
in  the  usual  way)  and  left  nothing  to  be  desired  as  regards  convenience  of 
manipulation.  It  proved,  however,  to  have  one  serious  defect:  it  lacked 
permanence.  The  makers  reported  that  they  had  some  difficulty  in  getting 
the  small  gelatin  patch  to  adhere  to  the  worked  glass  surface.  They  finally 
succeeded  so  well  that  after  rather  more  than  twelve  months'  use  (in  March 
1905)  the  patch,  contracting,  pulled  off  the  face  of  the  glass.  The  observa- 


METHODS  OF  OBSERVATION  AND   MEASUREMENT.  9 

tions  of  bright  stars  were  thus  interrupted,  but  not  before  several  series 
had  been  completed  whose  results  afford  a  convincing  proof  of  the  accuracy 
and  usefulness  of  the  method.  In  the  light  of  this  experience  it  appears 
that  color-screens  of  this  sort  should  either  be  made  of  some  less  perishable 
material  or  be  sealed  in  between  two  plates  of  glass.  A  complete  outfit 
should  consist  of  several  screens  of  graduated  degrees  of  absorption,  differing 
by  about  two  magnitudes.  With  such  an  equipment,  the  parallax  of  all 
stars,  even  the  brightest,  could  be  determined  photographically  with  equal 
accuracy  and  convenience. 

§5.  Other  Systematic  and  Quasi-Systematic  Errors. 

Systematic  error  may  also  arise  from  change  of  the  adjustments  of  the 
instrument  with  which  the  photographs  are  made.  For  example,  if  the  plate 
is  not  perpendicular  to  the  optical  axis  of  the  telescope,  the  images  near  the 
center  of  the  field  will  be  displaced,  relatively  to  those  at  the  edge,  to  an 
extent  depending  upon  the  amount  and  direction  of  the  tilt  of  the  plate. 
This  and  other  similar  adjustments  were  controlled  carefully  during  the 
progress  of  the  work  and  remained  satisfactory  throughout. 

We  may  now  pass  to  those  errors  which,  while  of  at  least  approximately 
random  character  for  different  plates,  affect  all  the  images  of  a  given  star 
on  one  plate  alike.  Perhaps  the  most  important  of  these  "plate-errors"  is 
that  called  "guiding  error"  by  Kapteyn.* 

If  the  clock  is  not  driving  correctly  it  is  probably  going  pretty  regularly 
fast  or  slow,  and  the  stars  on  the  plate  are  constantly  trailing  a  little  in  one 
direction  and  being  brought  back  by  the  action  of  the  control.  Under  these 
conditions  a  bright  star-image  is  displaced  relatively  to  a  faint  by  an  amount 
which  becomes  quite  sensible  before  the  distortion  of  the  disk  is  apparent 
on  inspection.  A  good  electric  control  will  correct  errors  as  soon  as  they 
amount  to  two  or  three  tenths  of  a  second  of  arc,  and  possibly  the  best  visual 
guiding  may  do  the  same.  The  automatic  control  has  this  advantage,  that 
one  can  set  the  clock  regulator  so  that  the  accelerating  and  retarding  trains 
come  into  operation  with  nearly  equal  frequency,  which  eliminates  guiding 
error,  properly  so  called.  A  slight  continuous  trail  in  one  direction  has  little 
or  no  effect  on  the  relative  position  of  the  star-images. 

When  two  plates  of  the  same  field  were  taken  on  the  same  night,  the 
adjustment  of  the  control  pendulum  was  usually  slightly  changed  between 
them  to  make  the  residual  influence  of  the  guiding  error  different  for  the  two. 

Distortion  of  the  gelatin  film,  so  far  as  it  affects  any  large  region  of  the 
plate,  is  eliminated  by  the  use  of  the  reseau  in  measurement.  All  that  need 
be  feared  is  local  distortion,  too  local  to  be  regarded  as  uniform  inside  a  single 
reseau-square  5  mm.  on  a  side.  A  bad  plate  of  this  sort  can  almost  certainly 
be  detected  by  irregularities  of  the  reseau  lines,  and  rejected  upon  mere 
inspection.  The  remedy  for  these  errors  is  to  separate  the  different  exposures 
well  and  not  to  take  too  many  on  one  plate. 

*Groningen  Pub.  No.  i,  p.  67. 


10  DETERMINATIONS   OF   STELLAR   PARALLAX. 

§6.  The  Photographic  Observations. 

The  photographs  were  taken  with  the  Sheepshanks  Equatorial  of  the 
Cambridge  Observatory,  a  coude"  telescope  of  the  polar  siderostat  type.  As 
the  instrument  is  of  unusual  design,  a  brief  description  is  appropriate  here.* 

The  main  tube  of  the  telescope  is  mounted  in  bearings  near  the  top  and 
at  the  bottom,  and  forms  the  polar  axis.  Toward  its  lower  end  it  carries 
the  declination  axis,  upon  which  turns  a  short  tube  carrying  the  object 
glass.  Upon  an  axis  concentric  with  the  declination  axis  is  borne  a  plane 
mirror,  which  is  geared  so  as  to  bisect  the  angle  between  the  polar  axis  and 
the  objective  tube.  The  light  of  any  star  toward  which  the  latter  is  directed 
is  thus,  after  reflection,  brought  to  a  focus  at  the  upper  end  of  the  polar  tube, 
which  passes  through  the  wall  of  the  building  in  which  the  telescope  is 
installed  into  a  closed  observing  room.  The  rest  of  the  telescope  is  pro- 
tected from  the  weather  by  a  light  cover,  which  is  moved  away  to  the 
southward  during  observations,  leaving  the  whole  sky  available,  except  for 
the  region  near  the  pole,  which  is  obscured  by  the  main  building. 

The  mounting,  which  was  constructed  by  Sir  Howard  Grubb,  is  very 
massive  and  stable,  owing  especially  to  the  position  of  the  eye-end  in  the 
axis  of  rotation.  It  is  possible  to  strike  the  eye-end  a  smart  blow  with  the 
hand  without  causing  any  displacement  of  the  guiding  star  perceptible  with 
a  high  power. 

The  object  glass,  by  Cooke  and  Son  of  York,  is  a  triple  photo-visual 
combination,  of  12.5  inches  aperture  (reduced  in  practice  to  12  inches)  and 
19.3  feet  focal  length.  It  is  practically  perfectly  achromatic.  The  image  of  a 
bright  star  is  quite  free  from  the  violet  glare  familiar  in  ordinary  instruments, 
and  its  spectrum,  observed  with  a  direct-vision  prism  over  the  eye-piece,  is 
linear  throughout  its  extent. 

The  plane  mirror,  18  inches  in  diameter,  was  figured  by  the  late  Dr. 
Common,  and  gives  very  perfect  definition.  As  an  illustration  of  this  it  may 
be  mentioned  that,  under  good  atmospheric  conditions,  such  double  stars  as 
OS  156  (o?6o)  and  OS  175  (o?65)  are  easily  divided,  the  dark  interval  being 
apparently  half  the  diameter  of  the  disks,  while  £  Bootis  was  repeatedly 
seen  elongated  (its  distance  at  the  time  being  oT35). 

The  driving  clock  and  electric  control  (of  the  Grubb  mouse-wheel  pat- 
tern) are  at  the  upper  end  of  the  polar  tube,  directly  under  the  eye  of  the 
observer— an  inestimable  advantage  in  practical  work.  The  latter  not  only 
corrects  errors  in  the  rate  of  the  driving  clock,  but  automatically  sets  it  right 
if  it  gets  more  than  a  few  hundredths  of  a  second  fast  or  slow,  compared  with 
the  controlling  pendulum.  The  rate  of  the  latter  can  be  varied  at  will,  even 
during  the  exposure,  to  correct  drift  of  the  guide-star  due  to  refraction,  etc. 

The  plate-holder  is  mounted  on  a  double-slide  carrier,  as  suggested  by 
Dr.  Common,  which  also  bears  a  guiding  eye-piece,  outside  the  plate,  which 

•Compare  the  description  by  Sir  Robert  Ball  in  Monthly  Notices,  Lix.pp.  152-155,  from  which  much 
of  the  present  account  is  taken. 


METHODS  OF   OBSERVATION  AND   MEASUREMENT.  II 

is  adjustable  in  both  coordinates  and  is  furnished  with  divided  scales,  so 
that  it  is  usually  easy  to  find  a  suitable  guide-star  with  any  given  object  at 
the  center  of  the  field  and,  once  the  scale-readings  for  this  are  recorded,  to 
set  the  telescope  again  on  the  same  center  within  a  few  seconds  of  arc.  Cor- 
rections to  the  guiding  in  either  coordinate  may  be  made  by  hand,  by  moving 
the  plate-holder  in  its  slides  by  the  screws  provided  for  the  purpose.  The 
performance  of  the  clock  and  control  was,  however,  so  uniformly  excellent 
that  no  attempt  was  made  to  guide  by  hand,  and  these  screws  were  only 
used  to  displace  the  plate  between  the  successive  exposures. 

The  plate-holder  itself  is  of  brass,  and  the  plate  (two  of  whose  edges  are 
ground  smooth)  is  held  in  place  by  springs,  against  metal  stops,  to  avoid 
possible  displacement.  The  proper  adjustment  of  the  plate-holder  with 
respect  to  the  telescope  is  assured  by  three  contact  pieces,  which  engage  with 
a  conical  hole,  a  slot  and  a  flat  surface  on  the  end-plate  of  the  telescope,  and 
is  maintained  by  strong  spring  clamps. 

The  focal  length  of  the  objective  varies  considerably  with  the  temper- 
ature, depending  not  only  upon  its  value  at  the  moment,  but  on  its  course  for 
some  time  previously.  It  was  therefore  necessary  to  determine  the  focal 
setting  for  every  evening  of  observation,  and  sometimes  to  change  it  during 
the  evening.  A  scale  attached  to  the  guiding  eye-piece  made  this  an  easy 
matter.  The  variations  in  the  scale-value  of  the  plates  are  mainly  due  to 
this  cause.  The  mirror  was  dismounted  and  re-silvered  three  times  during 
the  period  covered  by  the  observations,  on  July  27, 1904,  February  21,  1905, 
and  about  July  10, 1906.  There  is  no  evidence  that  this  affected  the  accuracy 
of  the  observations  in  any  way. 

The  adjustments  of  the  instrument,  and  especially  the  perpendicularity 
of  the  plate  to  the  optical  axis,  were  tested  from  time  to  time  and  remained 
throughout  satisfactory. 

The  plates  are  of  the  Barnet  "  Rocket "  brand  and  are  coated  on  "  patent- 
plate"  glass.  They  are  of  the  standard  Astrographic  size,  6.25  inches  square, 
which  with  the  Cambridge  telescope  gives  a  field  of  i°  28'. 

Four  exposures  were  usually  made  on  each  plate.  In  the  intervals  the 
plate  was  moved  0.5  mm.  by  means  of  the  screw  of  the  double-slide  plate 
carrier,  in  the  direction  of  the  y  axis  (declination),  except  for  a  few  double 
stars,  for  which,  to  avoid  confusion  of  images,  it  was  necessary  to  displace 
in  x,  or  in  both  coordinates.  The  length  of  the  exposures  varied  from  2  to 
10  minutes,  according  to  the  brightness  of  the  stars  under  investigation  and 
the  state  of  the  sky.  Most  of  the  exposures  were  of  4  or  5  minutes'  duration. 

A  standard  Gautier  reseau  (No.  88)  was  impressed  on  all  plates  before 
development,  using  divergent  light,  from  a  source  whose  optical  distance  from 
the  plate  was  equal  to  the  focal  length  of  the  telescope.  This  procedure  elimi- 
nates any  error  due  to  curvature  of  the  plate.* 

*Hinks,  Monthly  Notices,  LIX,  p.  532. 


12  DETERMINATIONS  OF  STELLAR  PARALLAX. 

The  developer  was  metol,  made  up  according  to  the  makers'  formula, 
and  development  was  continued  until  the  plates  began  to  show  traces  of  fog. 
They  were  fixed  in  a  simple  hypo-solution,  thoroughly  washed  in  running 
water,  and  dried  in  a  vertical  position,  with  the  x  axis  horizontal. 

With  the  normal  exposure  of  4  or  5  minutes,  the  faintest  stars  of  the  Bonn 
Durchmusterung  are  usually  visible  (though  not  well  measurable)  on  the 
plates.  On  fields  lying  in  or  near  the  Milky  Way,  many  more  stars  are 
shown  on  the  plates  than  appear  in  the  Bonn  Durchmusterung;  but  this 
is  not  usually  the  case  for  other  parts  of  the  sky. 

The  working  list  was  in  the  form  of  a  card  catalogue.  Each  star  has 
a  card  on  which  the  parallactic  ellipse  is  drawn  from  the  tables  given  by  Sir 
David  Gill  (Annals  of  the  Cape  Observatory,  vol.  vm).  On  the  ellipse  are 
marked  the  places  corresponding  to  the  dates  up  to  which  evening  observa- 
tions and  from  which  morning  observations  are  possible.  A  glance  at  the 
card  shows  whether  circumstances  are  favorable  or  unfavorable;  whether 
the  evening  observations  should  be  put  off  to  the  last  moment  or  may  be 
made  with  equal  advantage  any  time  in  the  preceding  month ;  and  whether 
the  morning  observations  must  be  got  immediately  after  the  earliest  possible 
date  or  may  be  delayed  without  damage.  The  conditions  vary  so  much  from 
star  to  star,  especially  in  a  latitude  like  that  of  Cambridge,  which  is  really  too 
high  for  convenient  parallax  work,  that  to  have  diagrams  always  in  sight  is 
really  necessary.  They  are  made  complete  by  a  tracing  from  the  Bonn  Durch- 
musterung chart  to  identify  faint  stars,  and  by  the  necessary  miscellaneous 
instructions. 

§7.  Measurement  of  the  Plates. 

Two  instruments  have  been  used  for  measuring  the  plates — the  measur- 
ing machines  of  the  Cambridge  Observatory  and  of  the  Princeton  University 
Observatory.  A  full  description  of  the  first  has  been  given  by  Mr.  Hinks  in 
Monthly  Notices,  LXI,  pp.  444-458.  The  second,  though  somewhat  simpler 
in  construction,  is  identical  with  it  in  all  essential  features,  which  alone 
need  be  described  here. 

The  star-image  is  referred  to  the  adjacent  lines  of  the  photographed 
r£seau.  Its  distances  from  these  are  measured  by  means  of  a  finely  divided 
glass  scale,  which  is  itself  movable  by  means  of  a  micrometer  screw. 

A  real  image  of  the  re"seau-square  with  its  contained  stars,  of  magnifica- 
tion unity,  is  formed  in  the  plane  of  the  scale  by  an  objective,  and  the  whole 
is  viewed  by  an  eye-piece  magnifying  20  diameters  (which  is  equivalent  to  a 
telescopic  magnifying  power  of  460) .  The  divisions  of  the  scale  are  0.05  mm. 
apart,  so  that  100  of  them  equal  the  r£seau-interval  of  5  mm.  The  scale 
alone,  therefore,  makes  it  possible  to  read  directly  the  distance  of  a  star 
from  the  r£seau  lines  to  one  one-hundredth  of  their  interval.  By  moving 
the  scale  by  means  of  the  screw  (which  has  a  pitch  of  0.5  mm.  and  a  head 
divided  into  100  parts)  it  is  possible  to  measure  the  distance  through  which 


METHODS  OF  OBSERVATION   AND  MEASUREMENT.  1$ 

the  scale  must  be  moved,  in  order  to  bring  first  a  re"seau  line,  and  later  the 
star-image,  into  the  middle  of  the  nearest  spaces  on  the  scale.*  The  excess 
of  the  distance  from  the  line  to  the  image  above  the  integral  number  of 
scale-divisions  between  these  two  spaces  can  thus  be  measured,  by  estimating 
tenths  of  a  division  on  the  micrometer  head,  to  0.0005  mm.  or  one  one- 
hundredth  of  a  scale-interval.  By  taking  readings  on  both  the  adjacent 
re'seau-lines,  the  error  of  "runs"  due  to  the  lack  of  exact  agreement  between 
a  r£seau-interval  and  100  scale-divisions  is  allowed  for,  and  the  coordinates 
are  thus  obtained  to  o.oooi  of  this  interval. 

Two  settings  were  usually  made  on  each  star-image  and  one  on  each 
of  the  adjacent  re'seau-lines.  The  accidental  error  of  setting  on  either  is 
so  small  that  it  is  not  usually  worth  while  to  make  more.  When,  however, 
the  first  two  settings  on  the  star-image  differed  by  more  than  0.0004  R 
(which  corresponds  to  0^07)  further  settings  were  made;  and  the  number 
of  settings  on  r£seau  and  image  was  doubled  for  the  "parallax  stars." 

The  whole  number  of  r£seau-intervals  is  read  on  scales  attached  to 
the  frame  in  which  the  plate  is  carried,  which  is  movable  so  that  any  desired 
r6seau-square  can  be  quickly  brought  into  the  field  of  the  microscope.  The 
x  coordinates  range  from  6  to  36,  and  the  y  coordinates  from  7  to  37.  The 
measured  star  coordinates  have  therefore  as  a  rule  six  significant  figures. 
This  numbering  was  adopted  so  that  the  optical  center  of  the  field  (which 
is  not  quite  at  the  geometrical  center)  might  have  the  coordinates  20,  20. 
The  images  of  the  star  under  special  investigation  were  always  made  to 
fall  near  the  latter  point. 
§8.  Errors  of  the  Measuring  Apparatus. 

It  is  clear  that  coordinates  thus  measured  are  liable  to  errors  of  several 
kinds,  arising  from  inaccuracies  in  the  re"seau,  scale,  or  screw,  from  optical 
distortion  in  the  measuring  machine,  and  from  personal  equation  of  the 
measurer.  Errors  of  the  r£seau  will  affect  the  determination  of  absolute 
places  of  their  stars  to  their  full  amount.  But  the  present  work  is  purely 
differential  and  all  the  images  of  a  given  star  are  not  only  in  a  given  re"seau- 
square,  but  in  the  same  part  of  it — their  coordinates  on  different  plates 
seldom  varying  by  as  much  as  one-fourth  of  a  re"seau-interval.  The  maximum 
influence  which  such  errors  can  have  on  the  deduced  parallaxes  or  proper- 
motions  is  therefore  but  a  small  fraction  of  the  division-errors  of  two  adjacent 
reseau-lines — which,  in  view  of  the  uniformly  high  accuracy  of  Gautier 
reseaux,  may  safely  be  neglected. 

The  micrometer  screw  is  used  only  to  measure  distances  of  the  order  of 
o.i  mm.  Its  progressive  errors  need  therefore  hardly  be  feared,  but  periodic 
errors  might  be  troublesome.  The  division  errors  of  the  scale,  on  the  con- 
trary, may  affect  the  measurements  with  their  full  value. 

The  errors  of  the  scale  and  screw  of  the  Cambridge  machine  have  been 
investigated  by  Mr.  Hinks.t  and  those  of  both  machines  independently  by 

*The  spaces,  not  the  divisions,  of  the  scale  are  numbered.       fMonthly  Notices,  LXI,  pp.  456-457. 


14  DETERMINATIONS  OF   STELLAR  PARALLAX. 

the  writer.  The  screws  (which  were  made  by  Messrs.  Brown  and  Sharpe,  of 
Philadelphia)  appear  in  both  cases  to  be  free,  from  sensible  errors,  either 
progressive  or  periodic. 

The  division  errors  of  the  Cambridge  machine  average  less  than  0.0005 
mm. — that  is,  less  than  the  least  reading  estimated  in  a  single  measurement— 
and  may  therefore  be  neglected. 

The  first  scale  furnished  with  the  Princeton  machine  was  much  less 
satisfactory,  one  end  being  seriously  in  error.  It  was  found,  however,  that 
if  this  end  was  not  used,  the  error  due  to  the  scale  would  on  the  average  be 
only  0.0004  mm.,  which  as  before  may  be  neglected.  The  scale  with  which 
the  makers  of  the  instrument  replaced  the  first  one  was  quite  satisfactory, 
its  errors  throughout  its  length  averaging  but  0.0003  mm-  As  different 
parts  of  the  scale  are  used,  almost  at  random,  in  setting  on  different  stars  and 
re"seau  lines,  the  division  errors  will  at  most  do  little  more  than  increase  the 
accidental  error  of  the  measured  coordinates. 

The  graduations  of  the  glass  scale  are  on  the  side  farthest  from  the  eye- 
piece. As  the  rays  from  the  plate  and  from  the  scale  have  identical  paths 
from  this  point  to  the  observer's  eye,  optical  distortion  in  passing  through 
the  glass  on  which  the  scale  is  ruled,  or  in  the  eye-piece,  can  not  affect  the 
measures,  but  optical  distortion  due  to  the  objective  which  forms  the  image 
of  the  plate  on  the  scale  may  do  so. 

This  distortion  of  the  field  was  investigated  for  both  machines  by  the 
writer,  by  measuring  the  distance  between  the  same  pair  of  re"seau  lines  in 
different  parts  of  the  field,  using  the  same  scale  divisions.  In  neither  case 
is  there  any  evidence  of  true  optical  distortion,  such  as  has  been  discovered  in 
machines  of  similar  type  at  Greenwich  and  Oxford.*  In  the  latter  machines 
the  objective  is  a  single  achromatic  lens,  while  in  the  others  it  is  a  doublet  (a 
"rapid  rectilinear"  camera  lens),  which  explains  their  freedom  from  error. 

In  the  Cambridge  machine,  however,  the  distance  between  the  two 
objects,  as  measured  by  the  scale,  appears  to  change  uniformly  as  the  objects 
are  moved  across  the  field  in  the  direction  of  the  line  joining  them.  This 
can  not  be  due  to  optical  distortion,  which  would  produce  effects  of  equal 
magnitude  at  equal  distances  on  each  side  of  the  optical  center.  Its  cause 
was  found  in  a  slight  tilt  of  the  glass  scale,  which  was  not  parallel  to  the 
focal  plane  of  the  objective.  The  divisions  of  that  part  of  the  scale  which 
is  nearest  the  objective,  if  a  scale  of  equal  parts  on  the  plate  was  projected 
on  them,  would  appear  too  long,  and  those  of  the  opposite  ends  too  short. 
This  error  is  uniformly  progressive  along  the  scale,  so  that  the  apparent 
lengths  of  the  (really  equal)  scale-divisions  form  an  arithmetical  progression, 
and  the  apparent  division  errors  of  the  scale  are  proportional  to  the  square 
of  the  distance  of  a  division  from  any  given  point  of  the  scale. 

It  is  easily  shown  that  the  influence  of  this  error  upon  the  measured 
coordinate  of  a  star-image,  which  has  been  referred  to  the  two  adjacent  re"seau 

•Monthly  Notices,  unv,  pp.  632,  643. 


METHODS  OF   OBSERVATION  AND  MEASUREMENT.  15 

lines,  correcting  for  "runs,"  is  independent  of  the  part  of  the  scale  used  and 
depends  only  on  the  position  of  the  star  relative  to  the  lines,  varying  as  the 
product  of  its  distances  from  the  two.  When  the  plate  is  turned  through 
1 80°  in  its  own  plane  and  remeasured  (which  is  necessary  for  other  reasons) 
the  scale-reading  for  the  star-image  is  affected  to  the  same  extent  and  in  the 
same  direction  as  before;  but  as  in  one  case  the  readings  increase,  and  in  the 
other  decrease,  with  increasing  coordinates  on  the  plate,  the  errors  in  the  meas- 
ured coordinates  will  be  equal  and  of  opposite  sign,  and  their  mean  will  be 
free  from  this  error.  It  is  therefore  of  no  practical  importance. 

The  Princeton  machine  is  free  from  sensible  error  of  this  sort — special 
care  having  been  taken  by  the  makers  (the  Cambridge  Scientific  Instrument 
Company)  to  avoid  it.  It  thus  appears  that  neither  measuring  machine  has 
any  errors  which  can  sensibly  influence  the  results  obtained  with  it,  and  there- 
fore no  corrections  for  instrumental  errors  are  necessary. 

It  is,  however,  well  known  that  most  measurers  have  a  systematic  ten- 
dency to  set  farther  to  one  side  when  bisecting  a  round  star-image  (or  when 
setting  it  between  two  wires  or  scale-divisions)  than  when  setting  similarly 
on  a  reseau  line.  This  error  may  vary  with  the  brightness  of  the  stars  and 
the  character  of  their  images,  and  it  is  universally  recognized  that  it  must  be 
got  rid  of,  as  far  as  possible,  by  turning  the  plate  through  180°  in  its  own 
plane  and  measuring  again.  In  the  case  of  symmetrical  images  at  least,  it 
seems  safe  to  assume  that  the  errors  in  the  measured  coordinates  will  be  equal 
and  opposite  in  sign  in  these  two  cases  and  that  the  mean  of  the  two  measures 
will  be  free  from  error. 
§9.  Economy  of  Measurement. 

At  first  all  the  images  were  measured  in  both  positions  of  the  plate,  as 
described  above.  But  later  it  was  suggested  by  Mr.  Hinks  that  if  this  error 
was  really  constant  for  images  of  a  given  intensity  and  appearance  it  should 
be  eliminated  from  the  mean  of  the  measures  of  several  similar  images  of  the 
same  star  by  measuring  half  of  them  in  one  position  only  of  the  plate  and 
the  rest  in  the  other.  Examination  of  the  measures  of  1304  star-images  on 
3 1  plates  showed  that  this  was  substantially  the  case.  The  mean  coordinates 
of  the  stars,  derived  from  measures  of  all  four  images  in  both  positions  of  the 
plate,  would  on  the  average  have  been  altered  by  only  0.00009  reseau-intervals 
(o''oi5)  if  the  first  two  images  had  been  measured  only  in  one  position  of 
the  plate  and  the  last  two  in  the  other.  This  is  less  than  one-third  of  the  aver- 
age probable  error  of  such  a  mean  coordinate.  It  follows  that  doubling  the 
labor  of  measurement  leads  to  only  a  small  increase  of  accuracy,*  which  does 
not  at  all  repay  the  additional  work.  The  rule  was  therefore  adopted : 

RULE  IV:  Measure  half  the  images  upon  each  plate  in  the  "  direct " 
position  in  the  machine  and  the  other  half  in  the  "reversed." 

When  there  are  but  three  measurable  exposures,  it  is  necessary  to 
measure  one  in  both  positions.  In  the  few  cases  where,  owing  to  gathering 

•For  further  details  see  Chapter  IV,  page  64. 


!6  DETERMINATIONS  OF   STELLAR  PARALLAX. 

haze  or  passing  clouds,  the  successive  images  of  the  same  star  were  not 
equally  intense,  care  should  be  taken  that  the  images  measured  in  opposite 
positions  should  be  as  nearly  comparable  as  possible.  The  work  of  measure- 
ment—one of  the  heaviest  parts  of  the  whole— was  thus  halved,  with  very 
little  loss  of  accuracy. 

It  is  possible  to  economize  measurement  still  further.  From  diagrams 
of  the  card  catalogue  it  appears  that  the  available  parallactic  displacement 
for  observations  on  the  meridian  is  rarely  more  than  half  as  great  in  the  y 
coordinate  (declination)  as  in  x,  so  that  for  parallactic  purposes  the  latter 
have  on  the  average  more  than  four  times  the  weight  of  the  former.  It  is 
therefore  desirable  to  confine  exact  measurements  to  the  x's  alone.  To  be 
sure,  we  need  approximate  values  of  the  y's  in  the  reduction  of  the  plates, 
but  these  can  be  obtained  from  rough  measures  of  a  single  plate — in  fact, 
of  a  single  exposure — in  perhaps  one-twentieth  of  the  time  that  it  would 
take  to  measure  the  y's  completely.  It  is  worth  while  to  measure  the  y's  on 
a  second  plate  belonging  to  the  epoch  farthest  removed  from  the  first,  as  a 
control,  and  in  order  to  detect  any  large  proper-motion  in  declination  among 
the  comparison  stars.  So  the  rule  was  adopted : 

RULE  V:  Measure  the  x's  accurately  on  all  plates,  but  the  y's 
approximately  on  two  plates  only. 

This  again  saves  half  the  work  of  measurement  and  reduces  it  to  a 
moderate  proportion  of  the  whole.  If  the  weight  of  the  y's  warrants  it  they 
may  be  measured  later,  in  the  few  cases  where  it  is  worth  while. 

By  making  the  plate-carrier  adjustable  in  position-angle  it  would  be 
practicable  to  set  the  x  axis  in  such  a  direction  for  each  star  that  the  whole 
available  parallactic  displacement  would  be  in  this  coordinate,  and  thus  get 
the  greatest  possible  return  as  regards  accuracy  for  a  given  amount  of  time 
spent  in  measurement.  This  was  unfortunately  impossible  in  the  case  of  the 
present  work. 

§10.  Working  List. 

The  stars  selected  for  observation  belong  to  two  classes: 

I.  Stars  for  which  any  information  as  to  the  parallax,  even  if  it 
be  only  a  superior  limit,  is  valuable,  such  as  visual  binaries, 
pairs  of  stars  with  common  proper-motion,  variable  stars,  and 
others  for  which  a  knowledge  of  the  parallax,  even  approx- 
imately, affords  information  of  astrophysical  value. 
II.  Stars  likely  to  have  larger  parallaxes  than  the  stars  in  general, 
such  as  those  of  large  proper-motion,  binaries  whose  apparent 
separation  is  great  in  proportion  to  their  period,  and  the  like. 
A  number  of  stars,  mostly  of  class  II,  were  also  included,  for  which 
the  investigations  of  previous  observers  gave  discordant  results,  or  which 
had  never  been  investigated  by  methods  of  the  highest  precision. 


METHODS  OF   OBSERVATION  AND   MEASUREMENT.  17 

The  working  lists  of  several  observers,  whose  results  have  appeared 
while  the  present  research  was  in  progress  (notably  the  great  series  of 
heliometer  determinations  at  the  Yale  Observatory),  were  independently 
constructed  on  much  the  same  plan.  In  consequence,  an  unusually  large 
proportion  of  the  stars  here  investigated  have  previously  been  observed 
elsewhere.  In  the  writer's  opinion,  this  is  by  no  means  to  be  regretted. 
The  present  state  of  the  problem  of  determining  stellar  parallax  is  such 
that  the  greatest  hope  of  advance,  as  well  as  the  best  test  of  the  absolute 
accuracy  of  the  work,  lies  in  the  comparison  of  results  obtained  by  as  many 
and  as  different  methods  as  are  capable  of  giving  satisfactory  precision; 
while  in  the  case  of  individual  stars  the  mean  result  of  short  series  of  obser- 
vations by  several  observers,  using  different  methods,  whose  agreement 
shows  them  to  be  free  from  serious  systematic  error,  is  entitled  to  more 
consideration  than  that  of  an  extended  and  elaborate  investigation  by  a 
single  observer. 

§11.  The  Magnitudes  and  Spectra. 

The  photometric  and  spectroscopic  data  contained  in  this  work  were 
obtained  at  the  Harvard  College  Observatory.  Prof.  E.  C.  Pickering — whose 
generosity  in  offering  this  entirely  unsolicited  and  very  valuable  information 
has  been  already  recorded — describes  the  methods  of  observation  as  follows : 

"  All  the  photometric  magnitudes,  measured  for  you  this  year,  were  obtained  with  the 
iz-inch  meridian  photometer.*  An  artificial  star,  formed  by  allowing  the  light  of  a  Wels- 
bach  burner  to  shine  through  a  small  hole,  is  reduced  to  equality  with  the  real  star  by  a 
wedge  of  shade  glass.  The  scale  is  reduced  to  that  of  the  4-inch  photometer,  by  measuring 
five  stars  taken  from  H.  A.  54,  before  and  after  each  of  your  groups.  The  magnitudes  of 
bright  stars  are  taken  from  H.  A.  50,  or  occasionally  from  H.  A.  54." 

[The  difference  of  magnitude  of  the  components  of  the  double  stars  in  the  list  was 
determined  from  measures  by  Mr.  Wendell.] 

"The  spectrum  of  each  star  was  estimated  by  Mrs.  Fleming  independently  on  two 
plates.  The  differences  are,  as  you  see,  insensible,  although  it  is  necessary  for  such  faint 
stars  to  use  spectra  only  about  2  mm.  long." 

In  some  cases  only  one  plate  of  the  region  was  available,  but  inde- 
pendent estimates  on  two  plates  are  recorded  for  164  stars.  For  134  of 
these  the  two  estimates  differ  by  less  than  half  the  interval  between  two 
adjacent  spectral  classes  (A,  F,  G,  K,  etc.).  In  25  cases  the  difference 
equals  half  a  unit, in  two  cases  0.8,  and  it  exceeds  one  unit  in  three  cases, 
in  all  of  which  one  of  the  estimates  is  recorded  as  doubtful.  It  is  therefore 
evident  that  the  probable  error  of  an  estimate  is  but  a  small  fraction  of  the 
interval  between  adjacent  classes.  For  this  reason  no  distinction  has 
been  made  in  the  tables  between  the  results  based  on  one  and  on  two  plates. 

•The  observer  being  Professor  Pickering  himself. 


CHAPTER  II. 

REDUCTION  OF  THE  MEASURES.* 

§  i  .  Formula  and  Standard. 

The  first  step  in  the  determination  of  the  star's  parallax  from  its  meas- 
ured coordinates  and  those  of  the  comparison-stars  is  to  reduce  the  rectangu- 
lar coodinates  measured  on  the  different  plates  to  some  uniform  standard. 
We  have  to  correct  not  only  for  those  causes  —  aberration,  refraction,  etc.  — 
which  alter  the  stars'  apparent  places  in  the  sky,  but  for  the  inevitable  errors 
of  centering,  scale-  value,  and  orientation  of  our  individual  plates.  It  is,  how- 
ever, a  well-known  advantage  of  working  in  rectangular  coordinates  that  all 
these  corrections  can  be  combined  into  one  very  simple  expression. 

It  was  first  shown  by  Turnerf  that,  except  for  photographs  taken  at 
great  zenith  distances,  all  the  necessary  corrections  are  sensibly  linear  func- 
tions of  the  measured  coordinates.  It  may  be  added  that,  for  plates  having 
the  same  center  among  the  stars  and  taken  at  a  fixed  hour-angle  (as  is  the 
case  here),  the  small  non-linear  terms  (which  arise  from  the  differential 
refraction)  are  practically  constant  for  each  star  and  affect  the  star-places 
alone,  not  the  parallaxes. 

If  then  x  and  y  are  the  measured  coordinates  of  a  star  on  any  plate,  and 
£  and  y  what  may  be  called  its  ideal  coordinates,  cleared  of  refraction, 
aberration,  etc.,  upon  a  plate  of  correct  orientation  and  predetermined  cen- 
tering and  scale-  value,  we  should  have  for  every  star  on  the  plate  relations  of 
the  form 


.     .     .  i)=y+dx+ey+f     ...         (i) 

where  a,  b,  c,  d,  e,  f,  are  constants  for  the  whole  plate. 

Owing  to  errors  of  the  measured  star-positions,  these  relations  will  not 
be  rigorously  true  ;  but  the  errors  are  in  practice  so  small  that  their  squares 
and  products,  and  also  their  products  by  the  plate-constants  a,  b,  d,  e  (which 
are  themselves  small),  may  be  neglected. 

Under  these  conditions  we  may  assume,  without  sensible  error,  that 
the  difference  of  the  measured  coordinates  of  the  same  star  on  any  two  plates 
(barring  errors  of  measurement  and  proper-motion)  is  a  linear  function  of  its 
coordinates  upon  either  one  of  them,  or  upon  any  other  plate  with  the  same 
center,  or  even  of  the  x  of  one  such  plate  and  the  y  of  another. 

It  might  seem  desirable  to  compute  standard  coordinates  for  our  stars 
from  their  catalogued  right  ascensions  and  declinations;  but  there  are  usually 
not  enough  catalogue  stars  on  the  plates;  the  errors  of  their  tabular  places 

*The  greater  part  of  this  chapter,  so  far  as  it  contains  new  material,  represents  the  unpublished 
investigation  of  the  writer,  referred  to  in  Monthly  Notices,  ixv,  p.  781. 
tMonthly  Notices  uv,  p.  n. 

19 


20  DETERMINATIONS  OF   STELLAR  PARALLAX. 

are  much  larger  than  those  of  the  photographs;  and  the  necessary  calcula- 
tions take  a  good  deal  of  time.  The  photographs  themselves  are  free  from  all 
these  objections.  But  a  standard  derived  from  them  may  differ  slightly  in 
scale  value  and  orientation  from  the  assumed  values  of  these  constants  which 
we  use  in  computing  the  parallax  factors  and  in  reducing  our  results  to 
seconds  of  arc. 

These  differences,  however,  are  in  practice  so  small  that  they  can  not 
sensibly  affect  the  deduced  value  of  the  parallax.  The  scale-value  never 
differs  by  as  much  as  one  part  in  a  thousand  from  the  assumed  constant 
(175^8  per  r^seau  interval).  This  will  affect  the  deduced  parallax  in  the 
same  proportion.  The  effect  of  error  in  orientation  is  to  change  the  parallax 
factor  in  x  by  an  amount  equal  to  the  corresponding  factor  in  y,  multiplied 
by  the  orientation  constant  b.  As  b  is  almost  always  less  than  o.oi  and  the 
parallactic  displacement  in  y  averages  less  than  half  that  in  x,  the  error 
so  introduced  will  be  at  most  about  one  two-hundredth  part  of  the  whole 
displacement.  Both  these  effects  may  therefore  be  neglected.  The  best 
procedure  is  therefore : 

RULE  VI :  Choose  any  plate,  or  the  mean  of  any  number  of  plates, 
as  a  standard,  and  reduce  the  others  to  this. 

It  is  not  even  necessary  that  the  x  and  y  coordinates  shall  be  derived 
from  the  same  plates. 

If  we  have  to  reduce  any  plate  to  the  standard,  we  must  assume  as  a 
first  approximation  that  our  comparison-stars  have  no  sensible  parallax  or 
proper-motion.  Each  of  them  gives  us  an  equation  of  condition  of  the  form: 

at+bi+c-(x-$=*v  (2) 

where  x,  y  are  the  coordinates  on  the  plate;  £,  rj  are  the  standard  coordinates; 
the  quantities  in  parentheses  are  the  observed  data;  and  v  represents  the 
inevitable  residuals.  These  equations  must  be  solved  for  the  plate-constants 
a,  b,  and  c. 

It  should  be  observed: 

1.  That  as  we  are  dealing  with  the  x's  only,  we  are  not  concerned  with 
such  debated  questions  as  the  identity  of  the  scale-value  of  our  plates  in  the 
two  coordinates,  and  no  a  priori  relations  can  be  assumed  between  the  three 
constants  which  have  to  be  determined. 

2.  That  by  expressing  x-£  as  a  linear  function  of  £,  77  rather  than  of  x,  y 
the  equations  of  condition  for  different  plates  differ  only  in  their  absolute 
terms. 

3.  That  the  values  of  77  need  be  known  only  approximately.    Since  b 
may  be  as  much  as  o.oi  they  should  be  carried  to  two  decimal  places  less 
than  those  of  x  and  £. 

4-  That  in  case  a  star  has  moved  sensibly  between  the  epochs  of  the 
plate  and  the  standard  we  must  correct  its  standard  coordinates  £,  T;  for  the 
amount  of  this  motion  before  calculating  the  expression  a£+br]+ c.  This 
correction  is  sensible  only  for  very  rapidly  moving  stars. 


REDUCTION   OF   THE   MEASURES.  21 

§2.  Dyson  s  Method  of  finding  the  Plate- Constants. 

These  equations  of  conditions  might  be  solved  by  least-squares.  But  by 
a  suitable  choice  of  comparison-stars  a  simpler  method  of  solution  may  be 
used  without  loss  of  accuracy.  This  method  was  first  described  by  Dyson,* 
and  proceeds  as  follows : 

Arrange  the  equations  of  condition  in  order  according  to  the  values  of  £ ; 
divide  them  into  two  groups,  one  with  large  and  one  with  small  £,  and  take 
the  mean  of  each  group. 

This  gives  two  equations  in  which  the  coefficients  of  a  are  considerably 
different  and  those  of  c  are  unity.  If  the  stars  are  distributed  over  the  plate 
with  tolerable  uniformity  the  mean  values  of  i\  for  the  first  two  groups,  and 
hence  the  coefficients  of  b,  will  be  nearly  the  same. 

Subtracting  the  second  of  these  from  the  first  we  obtain  an  equation  in 
which  the  coefficient  of  a  is  large,  that  of  b  is  small,  and  c  does  not  appear 
at  all.  Arranging  our  equations  now  according  to  the  values  of  77,  and  pro- 
ceeding similarly,  we  reach  an  equation  with  a  small  coefficient  of  a  and  a 
large  one  for  b.  These  two  equations  give  us  a  good  determination  of  a  and 
b.  The  value  of  c  can  then  be  found  by  substitution  in  any  one  of  the  four 
mean  equations  already  constructed.  All  four  will  give  the  same  value  of  c, 
as  they  are  not  independent  (since  from  either  the  first  pair  or  the  last  pair 
we  may  deduce  the  equation  obtained  by  adding  together  all  the  original 
equations  of  condition.) 

This  method  is  an  extension  of  the  familiar  one  of  forming  quasi-normal 
equations  by  altering  the  signs  of  the  equations  of  condition  so  that  all  the 
coefficients  of  a  given  unknown  are  positive,  and  then  adding.  Like  this,  it 
sometimes  fails  of  application.  For  example :  if  all  the  stars  on  a  plate  were 
confined  to  its  northeast  and  southwest  quarters,  large  £  and  ij  coordinates 
would  always  go  together,  and  vice  versa,  so  that  the  two  pairs  of  mean  equa- 
tions would  be  identical,  and  also  the  final  equations  between  a  and  b,  which 
could  not  be  determined  at  all  from  them.  If,  however,  such  a  plate  were 
divided  into  halves  by  its  diagonals  instead  of  by  parallels  to  its  sides,  and  the 
equations  of  condition  grouped  accordingly,  a  good  determination  of  a  and  b 
could  usually  still  be  made.  With  liberty  to  draw  the  dividing  lines  across 
the  plate  at  any  angle,  the  method  would  fail  only  when  all  the  stars  were  in 
one  straight  line;  and  in  that  case  the  least-square  solution  also  becomes 
illusory.  This  indicates  that  Dyson's  method,  so  modified,  ought  to  be  gen- 
erally applicable. 

§3.  Accuracy  of  the  Method. 

Treating  the  matter  analytically,  we  have  to  represent  our  observed 
quantities,  x-£,  by  a  linear  function  of  £,  ?;.  Instead  of  these  we  may  use 
any  rectangular  or  oblique  coordinates  p,  q,  on  the  plate,  represent  our  obser- 
vations by  a  linear  function  of  these,  and  transform  to  £,  TJ  when  necessary. 

•Monthly  Notices,  LV,  p.  61;  LVI.  pp.  118-119.    See  also  Turner,  Monthly  Notices,  uv,  p.  489;  Hinks, 
Astronomical  Journal,  No.  475. 


22  DETERMINATIONS  OF  STELLAR  PARALLAX. 

If  there  were  no  errors  of  observation,  either  Dyson's  method  or  that  of 
least-squares  would  lead  us  to  the  exact  expression  which  would  satisfy  all 
the  equations 

(3) 


We  have  therefore  to  concern  ourselves  only  with  the  effects  of  such  errors 
upon  the  plate-constants,  as  determined  by  the  two  methods. 

Suppose  that  we  have  n  equations  of  condition,  and  that  the  errors  of 
their  absolute  terms  are  A,  A,  ...  A..  In  applying  Dyson's  method 
we  divide  these  equations  into  two  groups.  Let  n,  and  n,  be  the  number  of 
equations  in  the  first  pair  of  groups,  and  let  the  sign  (/>),  denote  the  sum  of 
the  p's  for  all  the  n,  equations  of  group  i  ,  etc.  Similarly  let  n3  and  K4  be  the 
number  of  equations  in  the  second  pair  of  groups.  Then 


Finally  let  (p)  denote  the  sum  of  the  p's  for  all  n  eqt  citions.  Then  we 
shall  have  the  following  equations  to  determine  the  errors  da,  &b,  be  of  the 
plate-constants  : 


(P),  &»+(«),  »+n,  fc  =  (A),  (p\  &a+(q)4  &b+n4  dc=  (A)J 


These  four  equations  are  not  independent,  for  the  sum  of  either  the  first 
or  the  last  pair  gives 


It  follows  from  these  equations  that,  when  the  plate  has  been  reduced  to 
standard  by  this  method,  the  sum  of  the  outstanding  residuals  must  vanish 
for  each  of  the  four  groups;  and  hence  that  the  sums  of  the  residuals  for  the 
stars  in  each  of  the  four  quarters  into  which  the  dividing  lines  cut  the  plate 
must  be  numerically  the  same,  but  of  opposite  signs  in  adjacent  quarters 
(since  each  group  is  composed  of  two  adjacent  quarters).  This  affords  in 
practice  a  simple  and  complete  control  of  the  numerical  work. 

We  may  simplify  the  equations  (4)  by  a  proper  choice  of  axes.  Each 
of  the  four  groups  of  comparison-stars  has  a  centroid  whose  coordinates  are 
the  mean  of  those  of  the  stars  of  the  group.  If  the  p-axis  passes  through  the 
centroid  of  groups  3  and  4,  and  the  g-axis  through  those  of  groups  i  and  2,  the 
origin  will  fall  at  the  centroid  of  the  whole  system  of  comparison-stars.  The 
sum  of  the  p's  will  vanish  in  groups  3  and  4,  and  that  of  the  q's  in  groups  I 
and  2,  and  the  equations  (4)  will  become 

(p)lSa  +  n,Sc=(A).  (q\  Sb+n,6c  =  (AU 

(p),  la  +  n,  bc=  (A),  (fl)4  «6  +  n4  6c=  (A)J 

while  the  sum  of  either  pair  gives  simply 

(6) 


REDUCTION  OP  THE  MEASURES.  23 

If  the  probable  error  of  one  of  the  n  quantities  A  is  r,  that  of  8c  will  there- 

fore be  —  =,  whatever  the  choice  of  the  groups  1,2,3,  4-    Those  of  Sa  and  5b 
i'  n 

will  depend  upon  the  arrangement  of  the  stars  in  these  groups. 

It  is  usually  possible  to  choose  the  comparison-stars  and  group  them  so 
that  the  P-SLXIS  —  that  is,  the  line  joining  the  centroids  of  groups  3  and  4— 
completely  separates  the  stars  of  groups  i  and  2,  and  vice  versa.  In  this 
case  p  is  negative  for  all  the  stars  in  group  i,  and  positive  for  all  those  in 
group  2  ;  and  the  other  two  groups  have  the  same  property  with  regard  to  q. 

If  then  P  denotes  the  mean,  without  regard  to  sign,  of  the  ^-coordinates 
of  all  the  stars,  we  shall  have 

(P),=  -(P)>  =  ~  (7) 

Introducing  this  into  (5)  and  eliminating  Sc,  we  find 

p 


The  probable  error  of  the  second  member  is  r  V  n\n?-\-  n\nv-     Since  n,  +  n,  —  n 
this  reduces  to  r  V  n  ».«,,  and  the  probable  error  of  Sa  is 


(» 


But  4«,  n2  is  at  most  equal  to  n3.     Hence  we  have 


Proceeding  similarly,  we  find  rb<   —=.,  where  Q  is  the  mean  of  the  abso- 

—  Qyn 
lute  values  of  q  for  all  the  stars. 

We  may  now  compare  these  with  the  results  of  the  least-square  solution. 
Using  the  same  notation,  but  capital  letters  to  distinguish  the  plate-constants 
found  by  this  method,  the  normal  equations  are 

(P')8A  +  (pq)8B  =  (p£)         (pq)8A  +  (q')6B=*(q&)         w5c=(A)         (10) 

The  last  equation  is  identical  with  that  for  Sc  in  Dyson's  method.    The  two 
methods  therefore  give  identical  values  of  c. 

The  values  of  a  and  b  given  by  the  two  methods  will  in  general  be  differ- 
ent. The  probable  errors  of  the  constants  determined  by  least-squares  will 
be  given  by  the  equations 

(g')r' 
A  '  ' 


(/")(«)'  -to)' 


24  DETERMINATIONS   OP   STELLAR  PARALLAX. 

Since  the  sums  of  the  values  of  p,  corresponding  to  positive  and  negative 
values  of  q,  separately  vanish  (and  vice  versa)  the  summation  (pq)  will  be 
small  compared  with  the  others,  and  we  shall  have 


the  actual  values  usually  approaching  closely  to  equality.     Now,  unless 
all  the  p's  are  numerically  equal  (p*)>nP*.     We  may  set  (p')=KnP\ 

where  K>i.     If  the  stars  are  uniformly  distributed  over  the  plate,  K  =  -. 

O 

They  can  hardly  in  practice  be  so  irregularly  grouped  that  K  is  greater  than  2. 

We  then  have  rA  ^  —  r-^,  whence  by  (9)  ra  g  rkV^K,  and  similarly  r0  ^  ra  i7  K. 
Pi'nK 

The  diminution  in  the  probable  errors  of  these  plate-constants,  upon 
passing  from  the  approximate  method  to  that  of  least-squares,  is  therefore 
usually  small. 

For  the  actual  difference  between  the  two  values  of  a,  we  have,  in  the 
case  where  (pq)  =  o 


The  most  probable  value  of  this  difference  is  obviously  zero.  Its  probable 
error  R*  may  be  obtained  by  squaring  the  second  member  of  (12),  substitut- 
ing r'  for  the  square  of  each  of  the  A's,  and  zero  for  all  product  terms.  We 
thus  obtain 


r*       ^n  t,      . 

"1"1  «'  '' 


n'P(p')       n*P(p>) 
By  means  of  (7),  (8),  and  (n)  this  reduces  to 


(13) 


When  (pq)  is  not  zero,  the  same  final  equation  is  reached,  after  a  slightly 
longer  reckoning. 

Now  it  has  already  been  shown  that  K—  i  is  usually  considerably  less 
than  unity.  The  equation  (13)  may  therefore  be  stated  verbally  as  follows: 

The  probable  differences  between  the  values  of  a  and  b  obtained  by 
Dyson's  method  and  by  least-squares  (i.  e.,  the  limits  which  these 
differences  are  as  likely  as  not  to  exceed)  are,  under  all  ordinary 
circumstances,  considerably  less  than  the  probable  errors  of  the 
latter. 

It  remains  to  consider  the  probable  error  of  the  calculated  correction 
which  must  be  applied  to  the  measured  coordinates  to  reduce  them  to  the 
standard  system.  This  correction  is  ap+bq+c.  Since  the  three  plate- 


REDUCTION  OP  THE  MEASURES.  25 

constants  have  been  determined  independently  of  one  another,  the  probable 
error  r,  of  this  expression  will  be  given  by  the  equation 


The  correction  is  most  accurately  determined  for  a  point  at  the  centroid  of 
the  comparison-stars.  They  should  therefore  be  so  chosen  that  this  point 
falls  as  near  as  possible  to  the  parallax-star  which  is  the  main  object  of  the 
investigation.*  It  is  almost  always  possible  to  choose  them  so  that  for  the 

parallax  star  -^  and  ~-  are  less  than  0.3,  and  they  can  usually  be  made  very 

much  less.  The  uncertainties  of  a  and  b  then  contribute  less  than  one-sixth 
of  the  whole  uncertainty  of  the  calculated  correction.  These  are  the  only 
quantities  which  could  be  more  accurately  found  by  a  least-squares  solution, 
and  this  would  usually  diminish  their  uncertainty  by  only  about  one-third, 
and  hence  of  the  whole  correction  to  standard  by  only  about  one-twentieth 
of  its  whole  amount. 

The  error  of  the  final  reduced  coordinate  of  the  parallax  star  will  be 
the  resultant  of  the  error  of  its  own  measured  coordinate  and  of  the  correction 
just  discussed.  If  the  accuracy  of  the  measures  of  the  comparison-stars  is 
comparable  with  that  of  the  parallax  star,  it  is  clear  from  (14)  that  the  latter 
will  be  much  more  accurately  determined  than  the  former  —  its  relative 
weight  being  proportional  to  the  number  of  comparison-stars.  As  in  the 
present  work  there  are  always  at  least  five  of  the  latter,  it  is  clear  that  the 
uncertainty  of  this  correction  will  contribute  less  than  one-fifth  of  the  whole 
uncertainty  of  a  reduced  coordinate  (allowing  something  for  the  inferior 
accuracy  of  the  measures  of  the  comparison-stars). 

To  determine  the  plate-constants  by  least-squares  would  therefore 
increase  the  weight  of  the  reduced  coordinates  of  the  parallax-star  by  less 
than  i  per  cent,  which  would  be  far  from  repaying  the  additional  labor. 
The  rule  has  therefore  been  adopted: 

RULE  VII  :  Choose  the  comparison-stars  so  that  their  centroid  falls 
as  near  as  possible  to  the  parallax-star,  and  use  Dyson's  method 
of  reduction. 

In  dividing  the  comparison  -stars  into  groups,  in  applying  this  method, 
the  principles  laid  down  on  pp.  21,  23  must  be  observed. 

§4.  Number  of  Comparison-Stars. 

The  result  just  obtained  shows  that  very  little  accuracy  is  gained  by 
increasing  the  number  of  comparison-stars,  so  far  as  the  accidental  errors  of 
observation  are  concerned. 

If  we  wish  to  estimate  the  number  n  of  comparison-stars  which  will 
give  the  most  favorable  relation  between  the  work  expended  on  a  plate 

•Plummer,  Monthly  Notices,  LXIV,  p.  646. 


26  DETERMINATIONS  OF  STELLAR  PARALLAX. 

and  the  weight  of  the  resulting  coordinate  of  the  parallax  star,  we  must 
express  both  these  quantities  in  terms  of  n. 

If  r0  is  the  probable  error  of  a  measured  coordinate  for  the  comparison- 
stars  and  r,  that  for  the  parallax-star,  the  square  of  the  probable  error  of  the 

ri  /       p*     (f\ 
reduced  coordinate  of  the  latter  will  be  r*+^\l+~p*+Q>)  and  its  weight 

will  be  inversely  proportional  to  this  expression. 

The  work  of  taking,  measuring,  and  reducing  the  plates  may  be  divided 
into  two  parts,  one  of  which  is  independent  of  n,  and  the  other  proportional 
to  it.  It  may  therefore  be  set  proportional  to  n+k.  The  value  of  k  can 
only  be  roughly  estimated,  on  a  basis  of  experience:  for  the  present  work 
it  seems  certain  that  it  is  not  much  greater  than  8.  The  ratio  ^^t  is 
therefore  proportional  to 


This  expression  is  a  minimum  when 

<\ 

From  the  values  given  in  Chapter  IV  it  appears  that  the  second  factor 
in  this  expression  averages  about  1.7.  The  third  is  only  slightly  greater 
than  unity.  The  most  favorable  value  of  n  will  therefore  usually  be  4. 
This  reasoning,  however,  tacitly  assumes  that  it  is  always  as  easy  to  take 
more  plates  at  a  given  epoch  as  to  measure  more  stars  on  the  plates — which 
is  very  far  from  the  truth.  There  is  also  the  objection  that  with  so  few 
comparison-stars  it  is  impossible  to  detect  a  large  parallax  or  proper-motion 
among  them. 

Suppose  that  one  of  them  has  moved,  since  the  epoch  of  the  standard 
plate,  by  an  amount  which  is  large  compared  with  the  ordinary  errors  of 
observation.  If  there  are  but  three  comparison-stars,  there  will  be  only 
three  equations  of  condition  for  a,  b,  and  c.  The  calculated  values  of  these 
constants  will  satisfy  them  exactly,  leaving  no  residuals,  and  the  motion  can 
not  be  detected;  nor  can  unusually  large  errors  of  observation,  or  even 
numerical  mistakes  in  the  coordinates  to  be  reduced.  If  there  were  four, 
and  the  reduction  is  made  by  Dyson's  method,  the  residuals  for  all  four  stars 
will  be  numerically  equal  (two  positive  and  two  negative),  so  that  we  can  not 
pick  out  the  moving  star.  Something  of  the  same  sort  happens  whenever 
the  moving  star  is  alone  in  the  quarter  of  the  plate  where  it  lies. 

It  is  therefore  desirable  to  have  two  comparison-stars  in  each  quarter 
of  the  plate,  or  eight  in  all.  This  also  makes  it  much  easier  to  satisfy  the 
condition  of  Rule  VII  concerning  their  centroid.  The  usual  number  of 
comparison-stars  chosen  in  the  present  work  is  eight.  In  a  few  cases  it 
was  necessary  to  reduce  this  number  to  seven  or  six,  and  in  one  exceptional 
instance  to  five,  while  in  a  few  others  it  was  increased  to  nine  or  ten. 


REDUCTION  OP  THE  MEASURES.  27 

For  eight  comparison-stars,  setting  k  =  8  in  the  formula  above,  we  find 
that  about  one-eighth  more  work  must  be  expended  in  order  to  get  results 
of  equal  weight  for  the  parallax-star  than  if  there  were  only  four.  But  the 
certainty  that  the  results  are  not  modified  by  the  existence  of  a  considerable 
parallax  in  one  of  the  comparison-stars  compensates  for  this — to  say  nothing 
of  the  information  concerning  the  parallaxes  of  the  comparison-stars  them- 
selves that  can  be  deduced  from  their  residuals. 

§5.  Solution  for  the  Parallax  and  Proper-Motion. 

When  all  the  plates  of  a  field  have  been  reduced,  the  residuals  for  the 
parallax-star  are  converted  into  seconds  of  arc  (using  the  standard  value 
1 75  ".8  for  the  reseau  interval)  reduced  to  a  common  epoch  with  the  cata- 
logued proper-motion  of  the  star  (or  if  need  be  with  an  approximate  value 
deduced  from  the  plates  themselves)  and  discussed  by  least-squares  in  the 
ordinary  fashion,  the  unknowns  being  the  star's  parallax  and  corrections 
to  its  assumed  x  coordinate  and  proper-motion  in  x. 

It  is  not  worthwhile  to  make  such  an  elaborate  discussion  of  the  residuals 
for  the  comparison-stars,  because  any  parallax  or  proper-motion  in  any  one 
of  them  will  produce  systematic  changes  in  the  calculated  plate-constants, 
and  hence  in  the  residuals  for  all  the  other  stars,  so  that  the  results  for  the 
different  stars  are  not  independent. 

By  taking  means  of  the  equations  of  condition  for  each  parallactic 
epoch  and  combining  these  with  due  regard  to  their  weights,  results  prac- 
tically identical  with  those  of  the  least-square  solution  can  be  obtained  in 
much  less  time.  The  parallaxes  and  proper-motions  so  computed  for  the 
comparison-stars  will  usually  be  almost  wholly  due  to  errors  of  observation. 
If  any  star  has  a  large  one  this  can  be  at  once  detected.  Such  a  star  should 
be  rejected  as  a  comparison-star. 

It  is  a  good  illustration  of  the  advantages  of  photographic  work  in 
rectangular  coordinates  that  this  can  be  done  very  easily  indeed.  Owing 
to  the  linear  character  of  all  the  equations  used  in  reducing  the  plates  to 
standard  and  determining  the  parallax  and  proper-motion,  it  follows  that 
even  a  large  proper-motion  in  one  comparison-star  will  be  without  influence 
upon  the  computed  parallaxes  of  the  stars,  but  will  affect  only  their  calcu- 
lated proper-motions.  The  extent  of  this  influence  for  each  star  depends 
upon  its  position  on  the  plate,  and  also  in  some  degree  upon  the  way  in  which 
the  stars  are  divided  into  groups  in  using  Dyson's  method;  but  it  can  be 
readily  calculated  when  these  are  known.  We  can  then  determine  at  once 
what  proper-motion  we  must  assume  the  suspected  star  to  possess  in  order  to 
make  the  mean  of  those  of  the  other  stars  vanish  (that  is,  its  proper-motion 
relative  to  the  others,  considered  as  fixed)  and  also  those  of  all  the  other  stars 
referred  to  this  new  standard.  A  suspected  large  parallax  may  be  treated  in 
the  same  way.  In  neither  case  is  it  necessary  to  recompute  plate-constants 
and  the  like,  or  to  deal  with  any  quantities  saving  those  in  direct  need  of 


28  DETERMINATIONS  OF  STELLAR  PARALLAX. 

correction.  The  experience  of  the  present  work  indicates  that  such  cases  are 
rare  in  practice.  An  example  will  be  given  in  the  next  chapter. 

Even  when  the  computed  parallaxes  and  proper-motions  of  the  com- 
parison-stars appear  to  be  almost  wholly  due  to  errors  of  observation,  they 
furnish  information  of  much  value,  serving  to  detect  the  existence,  or  prove 
the  absence,  of  systematic  errors  —  depending,  for  example,  upon  a  star's 
position  on  the  plate,  or  upon  its  brightness  —  and  showing  also  how  the 
accuracy  of  the  measures  is  affected  by  these  conditions. 

If  the  weight  of  the  y  coordinates  justifies  their  being  discussed  at  all, 
they  may  be  handled  as  follows.  Choose  three  of  the  comparison-stars 
(already  found  to  have  no  sensible  proper-motion  or  parallax)  so  that  their 
centroid  falls  as  near  as  may  be  to  the  parallax-star.  Measure  the  y's  of 
these  four  stars  accurately  and  discuss  them  as  above.  The  reductions 
may  be  made  very  short.  If  the  subscripts  i,  2,  3,  a,  denote  quantities 
belonging  to  the  three  comparison-stars  and  the  parallax-star,  and  we 
determine  three  constants  a,  /3,  7  by  the  equations 


then  if  /  denotes  any  linear  function  of  £,  n 


The  correction  to  reduce  the  place  of  the  parallax-star  to  standard  may 
then  be  derived  immediately  from  the  differences  from  standard  for  the 
comparison-stars. 

The  parallactic  displacement  in  y  is  usually  so  small  that  even  this 
short  method  of  discussion  does  not  repay  the  labor. 


CHAPTER  III. 

THE  OBSERVATIONS. 

§i.  General  Summary. 

The  numerical  data  and  results  of  the  work  are  contained  in  table  C 
(at  the  end  of  the  volume).  Summarizing  them,  it  appears  that  the  present 
work  depends  on  254 plates,  of  37  different  fields;  of  these,  109  (43  percent) 
were  taken  by  Mr.  Hinks  and  145  (57  per  cent)  by  the  writer;  on  these  plates 
were  made  976  exposures,  of  an  average  length  of  4°  22s  and  an  aggregate 
duration  of  about  71  hours,  and  upon  them  were  measured  9232  images  of 
338  different  stars,  52  of  which  are  "parallax-stars,"  and  the  remaining  286 
comparison-stars.  The  parallax  was  determined  for  all  the  former  and  for 
242  of  the  latter,  excluding  only  those  in  the  incomplete  Series  xxxn  to 
xxxvir,  which  were  observed  at  two  epochs  only,  owing  to  the  accident  to 
the  color-screen  (see  page  8),  making  a  total  of  294  stars  whose  parallaxes 
are  given  in  this  work. 

§2.  Description  of  Table  C. 

The  observational  data  and  the  necessary  details  of  their  reduction  and 
discussion  are  given  in  table  C,  pages  104-142,  each  section  of  which  gives 
the  results  from  one  series  of  plates.  At  the  top  of  the  page  is  the  current 
number  of  this  series  and  those  which  the  star  or  stars  especially  observed 
for  parallax  bear  in  the  final  table  of  results;  then  follow  the  designations 
of  these  stars,  their  approximate  places  for  1900,  and  their  proper-motions — 
in  right-ascension,  declination,  and  on  a  great  circle.  The  latter  are  taken 
from  Boss's  Preliminary  General  Catalogue,  when  the  stars  appear  in  the 
latter;  otherwise,  usually  from  Bossert's  Catalogue  of  stars  of  large  proper- 
motion.  The  designations  of  the  stars  are  for  the  most  part  those  employed 
by  the  latter  authority. 

The  upper  half  of  each  table  gives  the  necessary  data  concerning  the 
individual  stars  and  plates,  and  also  the  measured  coordinates,  the  con- 
stants necessary  to  reduce  them  to  the  common  standard,  and  the  results  of 
this  reduction.  The  last  two  (or  occasionally  three)  also  give  proper-motions 
and  parallaxes  of  all  these  stars,  resulting  from  the  approximate  discussion 
described  below.  These  last  quantities  are  given  in  thousandths  of  a  second 
of  arc;  the  remainder  are  in  r£seau  intervals  of  175? 8. 

The  first  five  columns  of  the  table  deal  with  the  stars  observed.  The 
parallax-stars  are  denoted  by  letters,  and  the  comparison-stars  by  numbers 
(in  order  of  increasing  right  ascension),  which  are  given  in  the  first  column; 

the  second  gives  the  Durchmusterung  numbers  of  these  stars,  and  the  third 

29 


30  DETERMINATIONS  OF  STELLAR  PARALLAX. 

gives  their  magnitudes  from  the  same  source.  For  a  few  stars  which  do  not 
appear  in  the  Bonn  Durchmusterung,  the  second  column  is  left  blank  and  the 
third  contains  a  rough  estimate  of  magnitude  based  on  the  photographs. 

The  next  two  columns  contain  the  photometric  magnitudes  and  spectra, 
determined  at  the  Harvard  College  Observatory.  For  some  of  the  faintest 
stars,  and  a  few  others  which  lie  near  bright  ones,  these  data  are  not  avail- 
able and  the  corresponding  spaces  are  vacant. 

The  next  two  columns,  headed  "Standard,"  contain  the  system  of 
coordinates  of  these  stars,  used  as  the  standard  to  which  all  the  plates  were 
reduced.  The  coordinates  17  (given  to  three  decimal  places)  depend  upon 
measures  of  one  or  two  images  of  each  star  on  a  single  plate.  Their  probable 
error  is  less  than  a  unit  of  the  third  decimal  place,  as  is  shown  by  comparison 
with  similar  measures  upon  a  plate  of  the  last  epoch  of  each  series.  (These 
comparisons  were  undertaken  primarily  to  detect  any  possible  large  proper- 
motions  in  declination.  As  none  were  found,  it  is  needless  to  give  further 
details  here.) 

The  coordinates  £  (which  are  given  to  five  decimal  places)  are  sometimes 
derived  from  the  same  plate  as  the  TJ'S  and  sometimes  are  mean  values  including 
other  plates  as  well.  This  latter  policy  was  at  first  adopted  in  order  to  secure 
as  accurate  a  standard  as  possible;  but  it  was  afterwards  realized  that, 
owing  to  the  strictly  differential  character  of  the  work,  this  led  to  no  real 
advantage,  and  the  practice  was  discontinued. 

It  should  be  noted  that  where  the  ^-coordinates  are  the  mean  of  two 
or  more  plates  they  are  not  exactly  rectangular  with  the  ^-coordinates. 
As  the  latter  are  used  only  to  compute  the  reductions  to  standard,  this  is  of 
no  importance  for  the  purpose  in  hand.  If  a  set  of  rectangular  coordinates 
are  desired  for  any  purpose,  they  can  be  obtained  by  using,  with  the  tabulated 
TJ'S,  the  x's  of  the  corresponding  plate. 

At  the  bottom  of  these  columns  are  given  the  proper-motions  of  the 
parallax-star  or  stars,  in  reseau  intervals  per  year,  to  three  places  of  decimals. 
These  are  necessary  in  the  case  of  rapidly  moving  stars,  because  in  calcu- 
lating the  expression  a£-f-foj-fc,  used  in  reducing  any  plate  to  the  standard, 
the  values  of  £  and  17  should  be  those  for  the  date  of  observation.  These 
may  easily  be  computed  from  the  tabular  data,  and  this  correction  has  been 
made  whenever  its  effects  are  sensible. 

§3.  Data  for  Individual  Plates. 

The  following  double  columns  each  contain  the  data  for  one  plate.  At 
the  top  is  the  current  number,  from  the  observing  books;  next,  the  (astro- 
nomical) day  of  observation ;  then  the  number  of  exposures  (those  actually 
measured,  ignoring  any  defective  ones),  the  average  exposure-time  in  min- 
utes, the  hour-angle  of  the  principal  star  at  the  mean  of  the  times  of  these 
exposures,  and  finally  the  initial  of  the  observer  (H  =  Mr.  A.  R.  Hinks,  R  = 
the  writer). 


THE  OBSERVATIONS.  3! 

Then  follow  the  ^-coordinates  of  the  stars.  Only  the  decimal  part  is 
given,  as  the  whole  number  is  usually  identical  with  the  standard  £,  and, 
when  it  differs  by  a  unit,  the  true  value  is  obvious  upon  inspection  (since  the 
differences  from  standard  are  nearly  the  same  for  all  the  stars).  These  quan- 
tities are  the  means  of  the  x's  of  all  the  images  of  the  starwhichwere  measured 
on  the  plate.  The  measures  for  the  individual  images  are  carried  to  four 
decimal  places  only,  but  the  means  are  taken  to  five,  to  avoid  errors  due  to 
neglected  decimals  in  the  reduction. 

The  results  for  the  individual  images  are  not  given,  partly  because  of 
their  bulk,  but  mainly  because  (since  each  image  was  measured  in  but  one 
position  of  the  plate)  they  are  affected  by  the  personal  error  of  bisection 
(page  15)  which  is  only  eliminated  when  the  mean  is  taken. 

When  the  number  of  measurable  images  of  each  star  was  odd,  one  was 
measured  in  both  positions,  and  when  one  image  was  unlike  the  others  (e.g. 
fainter)  care  was  taken  that  this  should  be  the  one. 

In  certain  cases  the  images  of  the  parallax-star,  and  occasionally  of  other 
stars,  were  remeasured,  to  investigate  apparent  discordances.  In  almost  all 
cases  the  original  measures  were  closely  confirmed,  showing  that  the  trouble 
was  in  the  actual  position  of  the  image,  and  not  in  the  measures.  The  letter 
B,  at  the  foot  of  this  column,  denotes  that  all  the  images  were  measured  in 
both  positions  of  the  plate. 

It  is  next  necessary  to  explain  the  meaning  of  the  quantities  given  at  the 
foot  of  the  column  for  each  plate,  under  the  caption  "Average  residual." 
The  successive  exposures  on  one  plate  differ  of  course  in  centering.  They 
may  possibly  differ  also  in  orientation  (owing  to  changes  in  refraction,  etc.), 
but  they  must  be  practically  identical  in  scale-value.  If  the  orientation  is 
constant,  the  differences  between  the  x-coordinates  of  the  stars  for  a  given 
exposure  and  the  mean  x's  for  the  same  stars  for  all  the  exposures  should  be 
the  same  all  over  the  plate.  If  the  orientation  differs,  these  differences  should 
be  of  the  form  by+c. 

The  actual  differences  for  individual  images  will  not  agree  with  these 
theoretical  values,  owing  to  errors  of  measurement  and  of  the  real  position 
of  the  star  images;  and  the  residuals  obtained  by  comparing  them  with 
theory  afford  a  measure  of  these  errors. 

For  the  plates  first  measured,  the  assumption  of  differences  of  orienta- 
tion was  made,  and  the  residuals  determined  graphically.  The  results  did 
not  confirm  the  reality  of  such  differences;  and  later  in  the  work  changes 
of  centering  alone  were  supposed  to  exist.  The  residuals  found  on  the  first 
assumption  will  naturally  average  somewhat  less  than  on  the  second. 

The  average  value,  regardless  of  sign,  of  these  residuals  for  all  the 
star-images  on  each  plate  is  the  quantity  given  in  the  tables.  The  values 
resulting  from  the  graphical  process  are  distinguished  by  an  asterisk.  The 
calculation  of  these  residuals  gives  a  valuable  control  of  the  numerical  work 
up  to  this  point  and  serves  to  detect  any  serious  error  of  measurement  or 


32  DETERMINATIONS  OF  STELLAR  PARALLAX. 

any  grossly  bad  images.  Their  numerical  average  is  evidently  an  indication 
of  the  general  quality  of  the  plate. 

The  quantities  a,  b,  c,  at  the  foot  of  each  column,  are  the  plate-constants, 
determined  by  Dyson's  method.  If  x  is  the  measured  coordinate,  and  £  and  TJ 
the  standard  coordinates,  then  x  =  £+a£+6»j+c. 

These  constants  are  expressed  in  units  of  the  fifth  decimal  place  and 
are  given  with  accuracy  sufficient  to  insure  the  correctness  of  the  last  figure 
of  the  reduced  value  of  x.  It  should  be  observed  that  a  and  b  are  abstract 
numbers,  while  c  is  proportional  to  the  r6seau  interval.  Thus,  for  example, 
for  Plate  188  (Series  n)  we  have 

,x  =  £-f  0.0000968  £—  0.001875577+0.0556211 

The  four  groups  into  which  the  comparison-stars  are  divided  by  wide 
spacing  correspond  to  the  four  quarters  into  which  the  plate  is  divided  in 
using  Dyson's  method.  To  obtain  one  pair  of  the  quasi-normal  equations, 
take  means  for  the  stars  of  the  first  and  second,  and  for  those  of  the  third  and 
fourth  groups;  to  get  the  other  pair,  combine  the  first  and  the  fourth  groups, 
and  the  second  and  third.  For  Series  xv  the  stars  were  divided  into  three 
groups,  each  of  which  gives  directly  one  equation  between  the  plate-constants. 

The  column  following  the  measured  x's  gives  the  residuals  resulting 
from  their  reduction  to  standard,  in  the  sense  Plate  minus  Standard,  in  units 
of  the  fifth  decimal  place  of  a  r6seau  interval.  Negative  residuals  are  in 
boldface  type. 

For  the  comparison-stars,  the  sum  of  the  residuals  should  be  numerically 
equal  for  each  of  the  four  groups,  but  of  alternating  sign.  The  residuals  for 
the  parallax-stars  often  show  conspicuous  evidence  of  proper-motion,  even 
in  an  interval  of  a  few  months. 

In  two  cases  (plate  398,  Series  xin  and  plate  436,  Series  xvin)  the 
center  of  the  field  is,  through  some  error  in  setting,  at  some  distance  from 
its  usual  position.  In  these  cases  the  measured  coordinates  require  a  slight 
correction  for  the  effect  of  the  inclination  between  the  planes  of  the  plate  and 
of  the  standard. 

If  x,  y  are  the  coordinates  of  any  star,  referred  to  the  standard  plate- 
center;  X,  Y  those  of  the  center  of  the  plate  under  discussion;  and  F  the 
focal  length  expressed  in  rdseau  intervals,  it  is  easy  to  show  that,  if  the  cor- 
rections 

•,- 


are  applied  to  the  measured  coordinates,  the  results  will  be  connected  with 
the  standard  by  linear  relations  and  may  be  reduced  as  usual.  This  "non- 
linear" correction  is  given  immediately  after  the  measured  coordinates. 


THE  OBSERVATIONS.  33 

§4.  Approximate  Solution  for  all  Stars. 

The  last  two  (or  occasionally  three)  columns  of  the  upper  part  of  the 
table  ("Approximate  Solution")  give  the  values  of  the  proper-motion  ft, 
and  parallax  w,  of  all  the  stars,  as  derived  from  the  approximate  solution 
described  on  p.  27,  and  illustrated  below. 

When  one  comparison-star  gives  evidences  of  real  proper-motion,  the 
values  obtained  by  excluding  it  as  an  object  of  reference  (see  p.  27)  are  given 
in  the  third  column  headed  ft'.  Except  in  this  case,  the  computed  parallaxes 
and  proper -motions  of  the  comparison-stars  (being  linear  functions  of  the 
residuals  of  the  individual  plates)  must  have  the  same  properties  as  these 
with  regard  to  their  sum  by  groups.  The  values  for  the  parallax-stars  are 
included  to  test  the  accuracy  of  the  approximate  method  employed. 

§5.  Least-squares  Solution  for  the  Principal  Stars. 

The  lower  half  of  each  table  contains  the  least-squares  discussion  for 
the  parallax-stars.  In  this,  all  observational  quantities  are  expressed  in 
thousandths  of  a  second  of  arc.  The  value  of  the  reseau  interval  is  taken 
as  1 75'' 8 — which  was  derived  by  Mr.  Hinks,  in  the  course  of  reduction  of  the 
Cambridge  plates  of  Eros,  from  comparison  of  numerous  photographs  with 
meridian  places  of  the  stars. 

The  first  column  gives  the  number  of  the  plate;  the  second  (headed 
"Observed")  gives  the  residual  (Plate  minus  Standard)  for  the  star  in 
question,  and  the  third  the  correction  for  proper-motion  necessary  to  reduce 
each  observation  to  a  common  epoch.  This  epoch  is  given  at  the  head  of  the 
column,  and  the  assumed  value  of  the  annual  proper-motion  at  its  foot.  The 
latter  often  differs  slightly  from  the  more  accurate  values  at  the  top  of  the 
page,  which  were  not  available  when  the  reductions  were  begun. 

The  following  column  (headed  "Corrected")  gives  the  sum  of  the  quan- 
tities in  the  two  which  precede  it.  If  the  assumed  proper-motion  were 
correct,  and  the  star  had  no  parallax,  the  numbers  in  this  column  should  be 
identical.  An  assumed  value,  Ax  is  given  at  the  foot  of  the  column.  By 
subtracting  this  from  the  individual  entries,  we  obtain  the  absolute  terms  of 
the  equations  of  condition. 

When  there  is  more  than  one  parallax-star,  and  their  proper-motions 
are  different,  the  data  for  the  second  star  follow  those  for  the  first  (as  in 
Series  n  and  xxxi).  But  when,  as  is  more  often  the  case,  they  have  nearly 
or  quite  the  same  proper-motion,  the  observed  residuals  for  both  stars  are 
first  given,  then  the  proper-motion  correction  common  to  both,  and  lastly 
the  corrected  values  (as  in  Series  vn,  xv,  xix,  etc.).  When  the  assumed 
proper-motion  is  zero,  the  corrected  values  are  identical  with  the  observed, 
which  alone  are  given.  Following  them  come  the  equations  of  condition. 

These  are  in  the  form 

x  -\-ay-\-  pir  =  n 


34  DETERMINATIONS  OF  STELLAR  PARALLAX. 

where  x  represents  the  correction  to  the  assumed  value  of  A.r,  and  y  to  the 
assumed  proper-motion,  already  given,  TT  is  the  star's  parallax,  and  n  the 
observed  quantity.* 

The  coefficients  of  y  are  simply  the  interval  in  years  between  the  as- 
sumed epoch  and  the  date  of  observation.  Those  of  IT  (which  are  prac- 
tically the  parallax-factors  in  right  ascension)  are  computed  by  the  formula 

p  =  R  cos  D  sin  (.4  —  a) 

where  a  is  the  star's  right  ascension,  and  A ,  D,  and  R  are  the  sun's  right  ascen- 
sion, declination,  and  distance.  By  forming  a  table  of  log  (R  cos  D)  for  every 
five  days  throughout  the  year,  the  computation  was  made  very  simple.  Four- 
place  logarithms  were  used,  and  the  results  were  checked  by  calculation  in 
duplicate.! 

The  parallax-factors  in  y,  when  required,  were  computed  by  the  formula 

pj  =  R  sin  D  cos  6  —  R  cos  D  cos  (.4  —  a)  sin  8 

where  5  is  the  star's  declination. 

Following  the  equations  of  condition  are  the  residuals,  in  the  sense 
(0—  C),  derived  from  the  various  solutions.  At  the  foot  of  each  column  of 
residuals  is  the  weighted  sum  of  their  squares,  (pvv) 

The  last  column  gives  the  weights  assigned  to  the  equations.  Usually 
all  plates  were  given  unit  weight.  Some  poor  plates  were  assigned  weight  ]/i 
and  a  few  very  poor  ones  _%".  The  reasons  for  giving  low  weight  to  a  plate 
were:  (i)  small  number  of  exposures;  (2)  bad  observing  conditions  (accord- 
ing to  the  notes  in  the  observing  book) ;  (3)  bad  character  of  the  images  or 
rdseau  (noted  by  the  measurer  before  reduction);  (4)  discordance  of  the 
measures  of  the  different  exposures  on  the  plate  (shown  by  the  unusually 
large  "average  residual"). 

In  all,  29  plates  out  of  255,  or  about  one-ninth  of  the  whole,  received 
diminished  weight.  In  only  one  case  (Plate  207,  Series  v)  was  a  plate 
given  reduced  weight  a  posteriori  (because  of  discordance  appearing  upon 
reduction,  and  otherwise  unaccountable).  In  two  other  cases,  Plates  455, 
(Series  xxvn)  and  318  (Series  xxxvn),  images  of  one  star  upon  an  otherwise 
good  plate  were  entirely  rejected  for  similar  discordance. 

At  the  bottom  of  the  tables  are  the  normal  equations  and  their  solution. 
The  coefficients  of  the  former  have  been  checked  by  duplicate  computation. 
The  absolute  trrms  are  given  in  thousandths  of  a  second  of  arc. 

The  first  solution  gives  for  each  parallax-star  the  values  of  x,  y,  and  IT, 
with  their  probable  errors  and  weights,  and  also  r0,  the  probable  error  of  one 
equation  of  condition  of  unit  weight.  When  it  seemed  doubtful  whether  the 

•These  equations  are  written  with  the  observed  quantities  in  the  second  member,  in  conformity  with 
the  usage  of  other  works  on  stellar  parallax.  Compare  Cape  Annals,  vol.  vm,  part  II,  p.  1 1  n.  Yale  Trans- 
actions, vol.  II,  part  n,  p.  219. 

tErrors  in  the  parallax  factors  so  small  that  they  obviously  would  not  influence  the  deduced  results 
by  more  than  o'ooi  were  not  corrected. 


THE  OBSERVATIONS.  35 

computed  values  of  y  or  ?r  were  real,  additional  solutions  were  made,  in  which 
the  doubtful  quantities  were  assumed  to  be  zero,  or  occasionally  to  have 
some  value  otherwise  determined.  The  resulting  values  of  the  remaining 
unknowns,  the  residuals,  and  the  probable  errors  are  given  in  their  appro- 
priate places. 

For  two  series  (xi  and  xv)  the  y  coordinates  were  measured.  The  results 
are  given  as  Series  xia  and  xva.  They  are  arranged  similarly  to  the  other 
series,  except  that  the  data  already  given  in  the  previous  table  are  not  re- 
peated and  that,  instead  of  the  plate-constants,  the  three  reduction  constants 
a,  0,  7  are  given  (see  p.  28),  enabling  us  to  find  the  correction  to  reduce  star  A 
to  the  standard,  from  the  differences  from  standard  for  the  three  selected 
comparison-stars. 

The  y  coordinates  for  the  plate  formerly  chosen  as  the  standard  (e.g., 
Plate  191,  Series  xi)  differ  slightly  from  the  approximate  values  given  in  the 
preceding  table,  because  they  are  means  including  a  larger  number  of  expos- 
ures than  do  the  latter. 

§6.  Stars  Observed  at  but  Two  Epochs. 

The  stars  of  Series  xxxn  to  xxvn  are  those  for  which  observations 
were  interrupted  by  the  accident  to  the  color-screen  (see  p.  8),  so  that  photo- 
graphs could  be  obtained  at  only  two  parallactic  epochs.  For  all  of  these 
except  77  Geminorum  (Series  xxxn)  measurement  and  reduction  were  under 
way  before  the  series  were  cut  short,  and  the  results  are  presented  in  tables 
similar  to  those  for  the  majority  of  the  stars,  while  in  the  remaining  case  the 
rapid  reduction,  with  but  three  comparison-stars,  was  employed. 

The  reduction  of  the  residuals  to  a  common  epoch  and  the  equations 
of  condition  are  presented  just  as  in  the  preceding  tables. 

It  is  of  course  impossible  to  determine  the  proper-motion  and  parallax 
independently  from  these  equations.  The  course  adopted  has  been  to  form 
mean  equations  for  each  of  the  two  epochs,  and  from  these  to  determine  x 
and  T  in  terms  of  y  (the  correction  to  the  catalogued  proper-motion).  As 
these  are  all  stars  of  well-determined  proper-motion  the  terms  involving  y 
are  presumably  small. 

The  values  of  y  necessary  to  reduce  the  assumed  proper-motions  to  those 
of  Boss's  Preliminary  General  Catalogue  (which  appeared  after  the  first 
discussions  had  been  completed)  and  the  corresponding  values  of  TT  are  given 
at  the  bottom  of  the  table.  These  values  of  the  parallax  have  been  taken 
as  final. 

The  probable  error  of  one  plate  has  been  determined  as  usual;  that  of 
IT  by  dividing  the  probable  error  of  the  difference  of  the  two  epoch-means 
by  the  difference  of  the  parallax  factors.  As  it  can  be  shown  (see  Chapter  IV, 
p.  56)  that  there  are  no  sensible  systematic  errors,  this  process  must  give  a 
close  approximation  to  the  truth.  The  solutions  that  have  sometimes  been 
made,  assuming  TT  =  O,  serve  to  show  more  clearly  what  reliance  can  be 


36  DETERMINATIONS  OF  STELLAR  PARALLAX. 

placed  on  the  observed  parallaxes.  The  parallaxes  and  proper-motions  of 
the  comparison-stars  can  not  of  course  be  determined  for  these  series. 

At  the  bottom  of  the  page  are  found  certain  notes. 

The  "Observer's  Notes"  are  taken  from  the  observing  books  and  deal 
mainly  with  the  atmospheric  conditions  and  the  performance  of  the  driving 
clock  and  electric  control.  The  "Measurer's  Notes"  are  taken  from  the 
sheets  of  measures  and  deal  principally  with  the  character  of  the  images  and 
reseau  lines.  The  reasons  for  assigning  reduced  weight  to  any  plate  (other 
than  a  large  "average  residual")  can  usually  be  found  in  these  notes. 

Sundry  other  notes,  which  occasionally  appear,  are  self-explanatory. 

§7.  Success  in  Eliminating  Hour-angle  Error. 

It  is  desirable  to  extract  from  table  C  and  to  summarize  the  data 
which  show  the  degree  of  success  attained  in  eliminating  hour-angle  error. 
The  form  of  this  which  is  most  to  be  feared  in  photographic  work  is  atmo- 
spheric dispersion.  Its  influence  may  be  calculated  as  follows. 

Suppose  that  the  refraction  constant  /3  differs  by  6/3  for  two  stars  of 
different  spectral  types.  By  Turner's  well-known  formula,  the  increase 
of  the  *  coordinate  by  refraction  is  $X  (+  small  terms)  when  X  is  the 
coordinate  of  the  zenith  on  the  plate.  The  relative  displacement  of  the  two 
stars  is  therefore  8x  =  Xdfi 

For  a  plate  whose  center  is  in  declination  5,  taken  at  hour-angle  /,  in 
latitude  v-, 

y_ sin  t  cos  <f 

cos  8  cos  <p  cos  /  +  sin  5  sin  <p 

For  plates  taken  near  the  meridian,  /  is  small.  Expressing  it  in  minutes 
oj  time,  neglecting  its  square  and  higher  powers,  and  introducing  the  latitude 
of  Cambridge  (+52°!  2'),  this  equation  becomes 

X=  +0.0027*  sec  (&— <p) 

If  now  the  weighted  mean  hour-angles  at  which  morning  and  evening 
observations  of  the  stars  are  made  are  /,  and  /„  and  the  corresponding 
parallax  factors  in  x  are  p,  and  p,,  the  r^uluiig  error  Sir  in  the  derived 
relative  parallax  will  be 

X,-X, 

P,~P, 
whence 

Sir  t,—t, 

T^  =  +0.0027-      r  sec  ($-«>) 

dp  P.-p, 

If  the  observations  of  different  years  are  made  at  different  hour-angles, 
:  will  also  be  some  error  in  the  derived  proper-motion;    but  the  pre- 
ceding expression  for  the  influence  on  the  parallax  will  still  be  approximately 
correct. 


THE   OBSERVATIONS. 


37 


Table  3  gives  the  values  of  these  quantities  for  each  of  the  37  series 
of  plates. 

TABLE  3. 


Series. 

/,—  *£ 

Pi-Ps 

s 

Series. 

,,-, 

Pi-p, 

Sir 
dd 

Series. 

*-«, 

,,-,, 

6f> 

»» 

m 

m 

i 

2 

+       1 

+  4 

+  1.38 
45 

+O.OO2 
+O.OO8 

'4 
'5 

o 

-   5 

+  1.56 
49 

O.OOO 

—0.015 

26 
27 

+  12 

—  21 

+  1.24 
•34 

+0.029 
—0.046 

3 

-  6 

•51 

—  O.OI  1 

16 

+  3 

39 

+0.007 

28 

+  13 

•37 

+0.033 

4 

-   5 

•74 

—0.014 

17 

+  7 

.42 

+0.017 

29 

-  6 

.38 

—  0.012 

5 

~~  '3 

4' 

—0.026 

18 

+  8 

•39 

+0.018 

30 

+  8 

•36 

+O.O16 

6 

+   2 

39 

+0.004 

'9 

+  4 

.26 

+0.023 

31 

-  4 

•34 

—O.OI3 

7 

O 

49 

O.OOO 

20 

-   5 

•34 

—  O.OI2 

32 

o 

.78 

O.OOO 

8 
9 

—    2 
-23 

40 
•63 

—0.004 
—0.038 

21 
22 

o 

-  8 

39 
.07 

O.OOO 
—  O.O3I 

33 
34 

-  8 

+  i 

•75 
•35 

—0.013 
+0.003 

10 

—    1 

•52 

—  O.O02 

23 

-27 

•44 

—O.O5I 

35 

—  5 

.40 

—  O.OIO 

11 

+    I 

•50 

+O.O02 

24 

-25 

.  12 

-0.089 

36 

-  6 

.28 

—0.013 

12 

+21 

.46 

+0.039 

25 

-25 

.OI 

—  0.067 

37 

—   i 

.08 

—0.003 

13 

O 

•49 

O.OOO 

The  average  value  of  /,  - 12  regardless  of  sign,  is  7™  3,  and  that  of  pl—p2  is 
1.40.  It  follows  that  the  average  influence  of  the  hour-angle  error  upon 
the  parallaxes  should  be  equal  to  the  relative  displacement  of  the  star- 
images  due  to  a  change  of  5.2  minutes  in  hour-angle,  or  (since  most  of  the 
stars  are  in  considerable  north  declination)  an  actual  motion  of  the  telescope 
through  about  one  degree,  in  following  the  stars — which  is  almost  certain 

to  be  insensible.     The  average  value  of  — ,  without  regard  to  sign,  is  0.018; 

5/3 

with  regard  to  sign  it  is  —0.008. 

The  whole  change  in  /3,  from  the  visual  to  the  photographic  rays,  is 
about  o?8.  Between  any  two  stars  photographed  on  the  same  plate  it 
must  be  much  less.  For  61  Cygni — a  reddish  star  which  should  give  a  large 
value  of  5/3 — Bergstrand  finds  5/3  =  0*12  (mean  for  the  two  components).* 
If  it  is  equally  great  for  all  the  parallax-stars  the  average  error  of  one  parallax, 

due  to  this  cause,  would  be  0^002.     The  greatest  value  of  -^  is  —0.09,  for 

Series  24.    To  produce  an  error  of  oToi  in  the  parallax,  5/3  must  in  this 
case  be  o?  u .    It  is  not  likely  to  be  much  greater. 

For  the  stars  observed  with  the  color-screen,  it  is  uncertain  whether 
green  or  violet  light  did  most  to  make  the  photographic  image.  If  the  former 
alone  was  effective  5/3  would  be  about -o?  8.  For  these  stars  the  average 

value  of  — ,  regardless  of  sign,  is  o.oi  i,  so  that,  even  upon  the  most  unfavor- 
5/3 

able  hypothesis,  the  average  error  of  these  parallaxes  due  to  this  cause  would 
be  only  0^009. 

It  therefore  appears  that  the  residual  errors,  due  to  atmospheric  dis- 
persion, can  hardly  in  any  case  exceed  o?oi.  The  means  adopted  for  its 
elimination  have  therefore  been  successful.  What  they  have  cost,  in  loss  of 


*Astronomische  Nachrichten,  3999. 


TABLE  4. 


Mean 

Mean 

No.  of 

R.  A. 

/>>-*>.- 

series. 

2» 

•47 

8 

6 

•7* 

3 

10 

5' 

5 

U 

.38 

8 

18 

.21 

8 

22 

.36 

5 

3g  DETERMINATIONS  OF  STELLAR  PARALLAX. 

weight,  may  be  seen  from  table  4,  which  gives  the  mean  values  of  p,-p,, 
grouping  the  stars  according  to  their  right  ascension.  (The  first  group 
includes  all  stars  between  oh  and  4h,  &c.) 

The  available  parallactic  displacements,  actually 
realized,  as  here  shown,  average  about  12  per  cent  less 
than  the  theoretically  available  values  given  in  Chapter 
I,  page  5.  The  general  mean  for  all  the  series  is  1.40. 

By  removing  the  restriction  of  observations  to  the 
meridian  the  average  available  displacement  could  have 
been  increased  to  about  1.90  and  the  nominal  weight 
with  which  the  parallax  was  determined  nearly  doubled ; 
but  the  average  difference  between  the  hour-angles  at 
morning  and  evening  observations  would  have  been  more  than  three  hours, 
and  the  systematic  error  would  have  been  increased  thirty-fold. 

§  8.  Examples  of  Details  of  Measurement  and  Reduction. 

In  conclusion  it  may  be  well  to  give  examples  of  the  details  of  measure- 
ment and  reduction.  For  this  purpose  Series  n  may  be  selected.  Here 
there  are  nine  comparison-stars  and  two  parallax-stars — A,  originally  chosen 
on  account  of  its  large  proper-motion,  and  B,  added  because  it  happened  to 
appear  on  the  plates,  which  is  too  bright  to  give  images  of  the  best  quality. 
As  an  illustration  of  the  way  in  which  the  measures  were  made  and  recorded, 
we  may  take  Plate  351,  on  which,  according  to  the  usual  plan,  two  of  the 
four  exposures  were  measured  in  one  position  of  the  plate  and  two  in  the 

other. 

The  record  of  the  measures  of  the  first  star  (with  certain  explanations 

added  in  parenthesis)  is  as  follows : 

TABLE  5. 


Star  i. 

Exp.  i. 

Exp.  3. 

Exp.  3. 

Exp.  4- 

(Integral  part  of  x)  13 
(Integral  part  of  y)  21 

(Reseau)  17600 
(Star)        11298 
(Star)       i  i  302 
(Star) 

"7597 

11255 

11350 

1  I2(C 

8238 
14606 
14607 

8235 
14645 
14655 

14640 

Faint 

(Star)      

14648 

(Reseau)     7614 

76ll 

18235 

18240 

(Resulting  x). 

13.6309 

13.6353 

13.6370 

13.6409 

Here  the  first  line  gives  the  reading  on  the  re"seau-line  adjacent  to  the 
star  in  the  direction  of  decreasing  x  coordinates  on  the  plate  (which  in  the 
position  of  the  latter  for  the  measures  of  the  first  two  exposures  was  appar- 
ently that  of  increasing  readings  on  the  eye-piece  scale,  but  opposite  for 
the  last  two,  after  the  plate  had  been  reversed).  This  reading,  obtained  by 
adding  together  the  readings  of  the  scale  and  micrometer  screw,  is  given  in 
hundredths  of  a  scale-division. 


THE   OBSERVATIONS. 


The  last  line  gives  the  similar  reading  on  the  r6seau  line  adjacent  to  the 
star  in  the  opposite  direction.  These  two  readings  should  differ  by  just 
10,000  if  the  scale  stood  in  exactly  the  assumed  ratio  to  the  reseau-interval. 
The  differences  from  this  value  give  the  error  of  "runs,"  which  is  allowed 
for  in  the  usual  way. 

The  zero  from  which  these  readings  are  taken  is  an  arbitrary  one, 
depending  on  the  position  of  the  plate  at  the  moment  under  the  microscope. 
It  is  nearly  the  same  for  neighboring  images  measured  in  immediate  succes- 
sion, and  would  be  exactly  so  if  the  slides  in  which  the  plate-carrier  moves  were 
geometrically  perfect  (which  would  have  involved  a  quite  useless  expense). 
But  for  the  measures  in  the  reversed  position  of  the  plate  it  is  wholly  different. 

The  intervening  lines  give  the  similar  readings  on  the  star-images. 
Where  more  than  two  are  made,  the  reason  is  the  discordance  of  the  first  two. 
The  settings  on  this  star  are  unusually  discordant,  probably  because,  as 
noted  at  the  time,  it  was  faint.  The  differences  between  the  mean  of  the 
readings  on  the  image  and  that  on  the  first  r£seau  line,  corrected  for  "runs," 
gives  the  fractional  part  of  the  star's  x  coordinate.  The  integral  part, 
taken  from  the  setting  scales,  is  recorded  at  the  side  of  the  sheet. 

When  the  whole  plate  has  been  measured,  the  mean  of  the  coordinates 
of  the  four  images  of  each  star  is  taken.  These  are  given  in  table  C.  The 
differences  from  this  mean  are  then  formed  for  each  image  and  arranged  in 
columns  for  each  exposure;  the  mean  is  taken  for  each  column,  and  the  indi- 
vidual differences  are  used  to  determine  the  "average  residual,"  as  described 
above.  For  this  same  plate  the  resulting  table  is  as  follows: 

TABLE  6. 


Qi  „_ 

Exposure  i. 

Exposure  2. 

Exposure  3. 

Exposure  4. 

otar. 

d 

V 

d 

V 

d 

t> 

d 

V 

2 

-37 

+  12 

+  3 

+  11 

—  2 

-6 

+36 

-17 

3 

-44 

+  6 

—  12 

-  4 

O 

-4 

+56 

+   2 

-5« 

—    2 

_       *J 

+  i 

+  10 

+6 

+49 

-  4 

4 

-57 

-  7 

—2O 

—  12 

+11 

+7 

+66 

+  12 

5 

-54 

-  5 

-  9 

—    I 

+  5 

+  i 

+57 

+  4 

8 

-56 

-  6 

o 

+  8 

-   5 

-9 

+61 

+  7 

6 

-56 

-  7 

+  5 

+  13 

+  5 

+  i 

+47 

-  6 

7 

-44 

+  6 

-  4 

+  4 

—   I 

-5 

+49 

—  5 

9 

-46 

+  3 

-18 

—  10 

+  10 

+6 

+  54 

+  i 

A 

-48 

+   3 

-15 

—  7 

+  7 

+3 

+  56 

+   2 

B 

-53 

-  3 

-  9 

—   i 

+  3 

—  i 

+57 

+  4 

Mean 

-49  5 

-  7-9 

+  4-0 

+  53  5 

Sum  for  v 

59 

7* 

49 

64 

Average  residual  5.7. 

For  each  image,  d  denotes  the  difference  of  the  measured  x  from  the 
mean  for  the  star,  and  v  the  difference  of  this  from  the  mean  for  the  exposure. 
Both  are  expressed  in  units  of  the  fourth  decimal  place  of  a  reseau  interval. 
It  is  clear  that  any  serious  error  in  the  original  coordinates,  however  arising, 
can  not  escape  detection  in  such  a  table. 


40  DETERMINATIONS  OP  STELLAR  PARALLAX. 

The  sum  of  the  quantities  v,  either  by  rows  or  by  columns,  must  vanish 
(within  the  error  arising  from  neglected  decimals) .  Their  sums,  regardless  of 
sign,  are  given  at  the  foot  of  each  column  and  afford  a  measure  of  the  accuracy 
of  the  coordinates.  The  "average  residual,"  given  in  table  C,  is  the  numer- 
ical mean  of  the  values  of  v  for  the  whole  plate.  For  this  plate  it  is  about 
20  per  cent'greater  than  the  average,  showing  that  it  is  of  rather  poor  quality. 

Still  taking  the  same  plate  as  an  illustration,  each  star  gives  an  equation 
of  condition  for  the  plate-constants,  of  the  form  a£-\-bri+c  =  x  —  £.  For 
example:  star  2  gives  14.5780+9. 7386+c  =  -0*34302.  Taking  the  means 
of  these  for  the  first  two  and  the  last  two  groups  of  stars,  we  obtain 

14.8710+17.9656+^=  -0.31325 
25.5870+21.0986+*;=  —0.29142 
whence,  subtracting, 

10.7160+  3.1336       =+0.02183  (i) 

Similarly,  the  means  of  the  first  and  fourth  and  the  second  and  third  groups 
give 

21.9980+13. i04&+c  =  —0.32344 

19.3580+27.9586+^  =  —0.27323 
whence 

—  2.6400+14.8546     =+0.05021  (2) 

Solving  the  equations  (i)  and  (2)  we  have 

a  =  +0.0009969  6  =  +0.0035573 

and  substituting  in  the  four  original  equations,  we  find  c=  —0.39198.     The 
substitution  should  be  made  in  all  four,  to  detect  possible  errors  of  arithmetic. 
For  any  other  plate,  the  absolute  terms  only  of  equations  (i)  and  (2) 
will  be  different.     If  we  call  these  ct  and  ct  we  find  once  for  all 


TABLE  7. 


c=  +0.088709*;,— 0.0187  IOC2  6 

and  their  determination  becomes  very  convenient.  Having  found  the  plate- 
constants,  we  calculate  for  each  star  the  quantities  £+o£ +6ij+c.  The  excess 
of  the  observed  values  over  these  represents  the  combined  effect  of  errors 
and  of  the  star's  motion.  For  example,  for 
stars  A  and  B,  on  the  above  plate,  we  have 
the  results  shown  in  table  7. 

In  practice,  only  the  decimal  parts  are 
written  down.  In  carrying  out  the  numeri- 
cal work  of  reducing  the  plates  and  solving 
the  equations  resulting  from  them,  much  use 
was  made  of  an  arithmometer  at  Cambridge 
and  of  a  large  cylindrical  slide-rule  belonging 


A. 

B. 

1 

19.81751 

22.58960 

o£ 

+  0.01977 

-f-  0.02251 

br. 

+  0.07208 

-f-  0.05606 

C 

-  0.39198 

-  0.39198 

Sum 

19.51738 

22.27619 

Measured  x 

19.53016 

22.27571 

o—c 

+  0.01278 

—  0.00048 

to  the  Department  of  Civil  Engineering  at  Princeton.  The  latter  proved  to 
be  especially  convenient  in  solving  the  systems  of  linear  equations  which 
continually  present  themselves. 


THE  OBSERVATIONS.  4! 

§9.  Example  of  Approximate  Solution. 

As  an  example  of  the  approximate  solution  for  parallax  and  proper- 
motion,  we  may  take  Series  xxrr,  where  we  have  eleven  equations,  for  five 
parallactic  epochs.  Taking  means  for  each  epoch,  and  remembering  that 
Plate  496  has  half-weight,  we  have  the  equations 

Weight. 

x  —  o.68y+o.7i7r=  -f-     9  2 

3 

2 


—  o.6o7r=  —  768  2 

where  the  absolute  terms  are  those  for  star  A  and  are  expressed  in  units  of 
the  fifth  decimal  place  of  a  r6seau  interval,  and  where  y  represents  the 
whole  proper-motion  in  the  x  coordinate. 

Taking  the  means  of  the  first  two  and  the  last  three  of  these  equations, 
we  find 

x-o.59y+o.i<)ir=  —54  x+i.o8y+o.2Oir=  -585 

whence 

+  i.673f-|-o.oiT=  -531 

Taking  means  of  the  equations  in  which  the  coefficients  of  TT  are  of  the  same 
sign,  we  have 

x+o.3sy+o.6oir=  —326  x+o.S2y  —  o.tfw=  —443 

whence 

—  o.i  93>  +1.07*-  =  +117 
whence  we  find 


or  in  seconds  of  arc 

y-  - 
The  values  derived  from  the  least-square  solution  are 

y-—  0?554  ±oTo07  TT=  +0^095  =*=O? 

so  that  in  both  cases  the  agreement  is  well  within  the  probable  error  of  the 
latter  (definitive)  values. 

When  the  different  epoch-mean  equations  are  of  different  weight,  this 
fact  should  be  borne  in  mind  in  combining  them.  If  a  little  discretion  is 
shown  in  this,  the  results  of  the  approximate  solution  agree  very  closely  with 
those  obtained  by  least-squares.  This  is  a  consequence  of  the  approximate 
equality  of  all  the  positive  and  all  the  negative  coefficients  of  v  in  each  set  of 
equations  of  condition.  It  affords  a  valuable  control  on  the  arithmetical  work. 

§10.  Case  -when  a  Comparison-  Star  has  Sensible  Proper-Motion. 

It  remains  to  give  an  example  of  the  discussion  of  a  case  when  a  com- 
parison-star shows  evidences  of  proper-motion  (see  p.  27).  For  this  we  may 
take  Series  n,  where  the  computed  proper-motions  of  the  stars,  in  thousandths 
of  a  second  of  arc,  are  as  shown  in  table  8. 


DETERMINATIONS  OF   STELLAR  PARALLAX. 


TABLB  8. 


The  numerical  equality  of  the  sums  for  the  four  groups  (within  the 
errors  of  reckoning)  shows  that  the  arithmetical  work  is  correct;  but  the 
value  of  y  for  star  8  is  so  large  as  to  indicate  real  proper-motion  rather  than 
errors  of  observation.  This  motion  will  alter  all  this  star's  coordinates  except 
the  standard.  The  differences  x— £  will  be  affected,  and  hence  the  derived 
plate-constants  and  the  residuals,  not  only  for  star  8,  but  for  all  the  others 
as  well.  To  correct  all  these  would  be  very  laborious,  but  is  fortunately  quite 
unnecessary. 

Since  the  equations  for  the  reduction  of  all  the 
plates  are  linear,  and  differ  only  in  their  absolute 
terms,  a  change,  of  magnitude  z,  in  the  x  coordinate 
of  a  given  comparison-star  on  any  plate  will  produce 
changes  in  the  plate-constants  and  in  the  deduced 
residuals  for  any  star  (whether  this  or  another)  which 
bear  a  fixed  ratio  to  z. 

If  2  is  proportional  to  the  time,  the  changes  in 
the  residuals  for  all  the  stars  will  be  so;  that  is,  they 
will  be  indistinguishable  from  their  own  proper- 
motion.  Hence  a  real  proper-motion  in  any  compari- 
son-star will  introduce  spurious  alterations  into  the 
computed'proper-motions  of  all  the  stars  on  the  plate, 
but  will  be  without  influence  on  any  independent 
quantity  (e.g.,  their  computed  parallaxes). 

l<et  us  then  suppose  that  the  standard  x  of  star  8  is  changed  by  z,  while 
the  others  are  unaltered.  Applying  Dyson's  method  (as  above)  to  the  result- 
ing coordinates,  we  find  very  easily  the  plate  constants 

0=4-0.013062  6=4-0.019152  c=—  0.5382 

The  assumed  differences  from  standard  and  those  computed  by  means  of 
these  plate-constants  are  as  follows: 

TABLE  9. 


Sums 

Star. 

y 

by 

groups. 

2 

3 

-    15 

+    101 

}    +86 

i 

—       IO 

1         . 

4 

-     75 

/ 

8 

-  137 

+   220 

}    +83 

6 

+      65 

1 

7 

-    119 

-87 

9 

-     33 

J 

A 

+2794 

B 

-   106 

Star. 

Assumed. 

Computed. 

Residual  (A-C). 

2 

0 

—  0.162  1 

+o.i6az 

3 

0 

-0.1391 

+0.139* 

i 

o 

+0.067* 

—  0.067  z 

4 

0 

+0.2)4* 

-O.J34Z 

5 

0 

+0.3271 

-0.327* 

8 

* 

+0.374* 

+o.6a6z 

6 

0 

+0.1351 

-0.135* 

7 

o 

+0.0541 

-0.054* 

A9 

o 

0 

+0.111  * 
+0.101  Z 

—  O.I  I  1  I 
—  0.  101  Z 

B 

0 

+0.059Z 

—0.059* 

The  numbers  in  the  last  column  represent  the  spurious  proper-motions 
of  stars,  which  will  appear  as  a  result  of  the  calculations,  when  star  8  alone 
has  a  real  proper-motion,  of  magnitude  z.  Their  sums  by  groups  are  numeri- 
cally equal,  as  they  should  be. 


THE   OBSERVATIONS. 


43 


If  now  we  wish  to  find  the  proper-motions  of  our  stars,  referred  to  the 
other  comparison-stars  as  a  standard,  rejecting  star  8,  we  may  proceed  as 
follows  : 

The  rejection  of  star  8  leaves  star  5  alone  in  its  "group."  Had  we  deter- 
mined plate-constants,  etc.,  anew  by  Dyson's  method,  we  would  then  have 
the  resulting  proper-motion  for  this  star  equal  to  the  sum  of  those  for  stars 
2  and  3,  and  also  to  that  for  either  of  the  other  two  groups,  with  its  sign 
changed.  But  we  can  do  this  at  once  by  the  choice  of  a  suitable  value  of  z. 

Correcting  the  original  proper-motions  by  subtracting  from  them  the 
spurious  proper-motions  just  given,  we  find  for  star  5  the  value  — 137+0.3272, 
while  the  sums  for  the  other  three  groups  are 


Stars. 
2,    3 

i.  4 
6,  7.  9 


+  86  —  0.301  2 
-87  +  0.3012 
—  87+0.3002 


We  have  thus,  to  determine  2,  the  equation 

-137+0.3272= +86-0.3013 

whence  2  =+355. 

To  correct  our  individual  proper-motions  we  must  add  to  each  the  quan- 
tity given  in  the  third  column  of  table  9  (which  represents  the  influence  of  the 
change  of  plate-constants).  The  results  are  as  shown  in  table  10. 

TABLB  10. 


Star. 

2 

3 

i 

4 

5 

8 

6 

7 

9 

A 

B 

Previous  values  .  . 
Correction  

-15 
-58 

+  101 

—  10 
+24 

-75 

+8} 

-'37 
+  116 

+220 

+  m 

+  65 

+  48 

-119 

+  10 

-33 

4-2794 

4-    16 

-106 

+   21 

New  values 

—  7* 

+  « 

4-  u 

+  8 

—    2  1 

+3S1 

+  m 

4-    7 

4-28*0 

8c 

Sums  by  groups 

a 

+ 

22 



21 

+20 

excluding  No.  8. 

We  have  now  a  set  of  proper-motions  which  satisfy  (within  the  errors 
of  reckoning)  exactly  the  same  conditions  that  would  result  from  a  complete 
new  solution.  But  since  these  conditions  are  linear,  there  is  only  one  way  of 
satisfying  them.  Our  results  must  therefore  be  identical  with  those  of  the 
more  complicated  process.  They  are  given  in  table  C,  Series  u,  under  the 
heading  /it'. 

The  average  value,  regardless  of  sign,  of  the  computed  proper-motions 
of  the  remaining  comparison-stars  is  reduced,  by  the  exclusion  of  star  8, 
from  69  to  48,  while  the  resulting  value  for  the  proper-motion  of  star  8  comes 
out  more  than  three  times  as  large  as  any  of  the  others.  Its  reality  is  thus 
confirmed,  but  its  amount  is  not  great  enough  to  justify  the  rejection  of  this 
star  as  a  comparison-star  for  parallax. 


CHAPTER  IV. 


DISCUSSION  OF  THE  OBSERVATIONS. 

I.  ABSENCE  OF  SYSTEMATIC  ERRORS. 

§  i .  Errors  of  Observation  almost  wholly  Accidental  in  Character. 

In  the  course  of  the  reduction  of  the  31  series,  which  form  the  principal 
part  of  the  present  work,  the  parallaxes  of  242  comparison-stars  have  been 
determined.  If  any  systematic  errors  affect  the  results,  this  large  amount 
of  material  should  suffice  for  their  detection,  provided  that  it  is  true,  as  is 
now  generally  believed,  that  the  individual  differences  of  parallax  among  such 
stars  are  practically  insensible.  To  see  whether  this  is  the  case,  the  numbers 
of  observed  parallaxes  lying  between  different  numerical  limits  were  counted 
with  the  results  shown  in  table  1 1 . 

The  second  column  gives  the  number  TABLS  H. 

of  parallaxes  which  lie  between  the  given 
limits,  while  the  third  column  shows  the 
distribution  resulting  from  the  "law  of 
errors"  with  probable  error  ±0*0283. 

It  is  at  once  manifest  that  the  observed 
parallaxes  of  the  comparison-stars  are  almost 
wholly  due  to  errors  of  observation  and  that 
they  furnish  no  information  at  all  about  the 
real  parallax  of  individual  stars. 

But  this  is  just  the  condition  under 
which  they  are  suitable  for  the  investigation  of  systematic  errors.  Such 
errors  may  depend :  (a)  upon  the  star's  position  upon  the  plate,  (b)  upon 
its  brightness,  (c]  upon  its  spectral  type,  (d)  upon  the  season  of  the  year 
in  which  observations  are  made. 

§2.  Search  for  Systematic  Error  depending  on  position  on  the  Plate. 

To  investigate  possible  errors  of  class  (a)  a  diagram  was  prepared,  show- 
ing the  position  of  each  of  the  comparison-stars  in  the  field,  and  its  observed 
parallax.  The  field  was  then  divided  into  sixteen  regions,  containing  nearly 
the  same  number  of  stars,  by  means  of  the  reseau  lines  'x  =  20,  y  =  20,  which 
pass  through  the  optical  center,  and  the  lines  five  reseau  intervals  (H'S) 
distant  from  these  on  each  side. 

The  results  may  best  be  represented  graphically.  In  the  diagrams 
which  follow  the  ^-coordinates  increase  toward  the  left,  and  the  y  coordinates 
upward.  Diagram  A  shows  the  number  of  stars  in  each  region,  and  B  their 

45 


Limits. 

Observed. 

Theory. 

+0^12  to  +0.09 

4 

3 

+0.09  to  +0.06 

16 

15 

+0.06  to  +0.03 

4> 

+0.03  to     o.oo 

61 

6} 

o.oo  to  —0.03 

64 

63 

—0.03  to  —0.06 

36 

39 

—0.06  to  —0.09 

«4 

"5 

—O.O9  tO  —  O.I2 

2 

3 

<         —  O.I2 

3 

1 

46 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


average  parallax,  regardless  of  sign,  in  thousandths  of  a  second  of  arc.  Dia- 
gram C  gives  the  average  parallax,  taking  account  of  sign,  for  each  region,  and 
D  the  numerical  values  which  this  might  be  expected  to  have,  if  due  to 
accidental  errors  alone  (which  are  obtained  by  dividing  the  numbers  in  B  by 
the  square  root  of  the  number  of  stars  in  each  region,  given  in  A) — all  in 
thousandths  of  a  second  of  arc. 


A. 


B. 


c. 


D. 


9  18  19  17 

49  27  23  37 

+20  -  9  -  1  +14 

16   6  5   9 

20  11  18  13 

32  53  26  31 

-10  +15  -  2  -  1 

7  16  6   9 

20  13  11  15 

34  38  22  34 

+  12  -  5  -11  -  1 

8  11  7   9 

17  12  15  14 

35  31  34  39 

+13  -  9  -20  -  4 

8   9  9  10 

Of  the  sixteen  quantities  in  C,  seven  are  less  than  might  be  expected, 
one  is  equal  to  expectation,  and  eight  are  greater.  The  numerical  mean  of 
all  the  observed  quantities  is  0^0092,  and  of  those  predicted  by  the  theory 
of  errors  0^0090.  It  would  therefore  appear  that  the  observed  quantities  are 
almost  wholly  due  to  accidental  error. 

Taking  means  for  the  four  middle  regions,  the  four  corner  ones,  and  the 
eight  others  at  the  sides,  the  results  are  given  in  table  12: 

TABLE  12 — Mean  Parallax. 


Without 
regard 
to  sign. 

With 

regard 
to  sign. 

Expectation. 

No.  of  stars. 

Corners  

oToio 

-f-oroio 

Sides  

o  o*  i 

Middle  

O  OH 

—  o  ooi 

All  

O   (H}2 

The  fourth  column  gives  the  values  which  the  quantities  in  the  third 
column  might  be  expected  to  have  if  due  to  accidental  error  only.  It  appears 
that  there  is  no  systematic  error,  unless  perhaps  a  very  small  one  for  the 
stars  at  the  extreme  corners  of  the  field,  which  in  any  case  must  be  less  than 
one-hundredth  of  a  second  of  arc.  The  average  value  of  the  parallax, 
regardless  of  sign,  is  25  per  cent  greater  for  this  group  than  for  the  others, 
showing  that  in  this  region  the  measures  become  less  accurate. 

§3.  Search  for  Error  depending  on  Magnitude. 

To  investigate  possible  errors  of  type  (b),  depending  on  magnitude,  the 
stars  were  grouped  according  to  their  photometric  magnitudes  (a  few  stars 
which  were  not  observed  photometrically  being  distributed  among  the  groups 
on  the  basis  of  their  Bonn  Durchmusterung  magnitudes). 


DISCUSSION  OF   THE  OBSERVATIONS. 

The  results  are  as  follows: 

TABLE  13. 


47 


No.  of 
stars. 

Mean  magnitudes. 

Mean  parallax. 

Expectation. 

Photometric. 

Bonn  Durch- 
musterung. 

Without  re- 
gard to  sign. 

With  re- 
gard to  sign. 

25 
59 
106 

52 

7.33 
8.64 
9  51 
10.35* 

7  39 
8.55 
9.07 
9.46* 

0*037 
0.030 
0.034 
0.032 

+0*006 

—  O.OOI 
—  O.OO3 
+0.003 

0*007 
0.004 
0.003 
0.004 

Here  again  there  is  clearly  no  sensible  systematic  error.  The  observed 
mean  parallaxes  are  smaller  than  might  be  expected.  This  is  probably 
to  be  explained  by  the  uneven  distribution  of  stars  of  a  given  magnitude 
among  the  different  fields.  The  sum  of  the  parallaxes  of  all  the  comparison- 
stars  in  a  given  field  necessarily  vanishes;  and  if  all  of  them,  or  all  but  one 
or  two,  fall  in  the  same  magnitude-group  they  will  make  an  unduly  small 
contribution  to  the  total  for  this  group.  This  effect  should  be  least  for  the 
groups  which  contain  fewest  stars  (to  which  each  field  will  contribute  but 
one  or  two  members) ;  and  this  is  in  accord  with  the  facts. 

The  values  of  the  average  parallax,  regardless  of  sign,  show  that  the  meas- 
ures are  less  accurate  for  the  brightest  stars  than  for  the  others,  for  which  they 
are  almost  equally  good. 

§4.  Search  for  Error  depending  on  Spectral  Type. 

It  is  possible  in  the  present  work  to  investigate  the  errors  of  class  (c) 
(depending  on  the  spectral  type)  directly. 

The  whole  number  of  comparison-stars  whose  spectral  type  was  deter- 
mined at  Harvard  is  216.  These  are  divided  almost  equally  among  the  four 
principal  classes.  In  table  14  all  spectra  from  A6  to  FS,  inclusive,  are  counted 
as  F,  and  so  on,  except  that  the  three  stars  of  type  M  are  included  with  those 
of  type  K. 

TABLE  14. 


Spectrum. 

No.  of 
stars. 

Mean  photo- 
metric 
magnitude. 

Mean  parallax. 

Expectation. 

Without 
regard 
to  sign. 

With 
regard 
to  sign. 

A.. 

40 

11 
58 
26 

9.03 
9.48 
9.09 
8.85 
9.92 

0*031 
0.033 
0.033 
0.036 
0.033 

—0*005 
+0.008 
+0.003 

—  O.OO2 
—  O.OII 

0*005 
0.005 
0.004 
0.005 
0.006 

F    

G     . 

K 

Not  observed  . 

'Several  of  the  faintest  stars  do  not  appear  in  the  Bonn  Durchmusterung,  and  were  not  observed  at 
Harvard.  If  they  were  included,  the  mean  magnitudes  in  the  last  line  would  be  slightly  lower.  They  are 
included  in  the  mean  for  parallax. 


48 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


Once  more  it  is  clear  that  the  systematic  error  must  be  quite  insensible 
The  slight  deficiency  in  parallax  for  the  (very  faint)  stars  whose  spectra  were 
not  observed  is  not  confirmed  by  the  result  previously  found  for  the  whole  of 
the  faintest  stars,  and  it  is  doubtless  due  to  accidental  errors  of  observation. 

§5.  Search  for  Errors  depending  on  Right  Ascension. 

There  may  also  exist  systematic  errors  arising  from  change  in  the  instru- 
ments, or  in  the  other  conditions  of  observation,  from  time  to  time,  and  espe- 
cially with  the  seasons.  The  influence  of  such  errors  upon  a  star's  observed 
parallax  will  be  a  function  of  its  right  ascension.  In  studying  such  errors 
it  is  no  longer  legitimate  to  combine  stars  in  all  parts  of  the  sky,  as  has 
previously  been  done.  They  have  accordingly  been  divided  into  groups, 
covering  4h  in  right  ascension,  and  for  each  of  these  the  difference  of  the 
mean  parallaxes  has  been  determined :  (a)  for  stars  in  the  inner  and  outer 
parts  of  the  field ;  (b)  for  the  brighter  and  fainter  stars ;  and  (c)  for  those  of 
"earlier"  or  "later"  spectral  type  (which  may  temporarily  for  convenience  be 
called  "white"  and  "red"),  dividing  the  stars  of  each  field  into  groups  as 
nearly  equal  as  possible.  The  results  are  as  follows : 

TABLE  15. — Mean  difference  of  parallax. 


R.  A. 

Inner  minus  Outer. 

Bright  minus  Faint. 

White  minus  Red. 

Obs. 

Exp. 

Obs. 

Exp. 

Obs. 

Exp. 

0* 

f 

+0*001 

+O.OOJ 

oTooS 
0.009 

+o!oi8 

—0.001 

oTooS 
0.009 

+oToo8 
-0.008 

0^009 

O.OIO 

I3k 

|6» 

30* 

Average 

—0.003 
+0.005 

+O.OO3 

0.009 

O.OII 
O.OII 

+0.010 
-0.008 
—  0.016 

0.009 

O.OII 
O.OII 

+0.017 

+0.014 

—  O.OO) 

O.OIO 

0.014 

O.OII 

0.00) 

0.010 

O.OII 

0.010 

0.010 

O.OII 

TABLE  16 


The  expectation  has  been  calculated  as  usual.  It  is  somewhat  larger 
in  the  last  column,  because  fewer  stars  were  observed  for  spectrum  than  for 
magnitude  or  position.  The  last  line  contains  the  average,  regardless  of  sign, 
of  the  quantities  in  the  preceding  lines. 

It  is  clear  that  there  is  practically  no  systematic  error  of  any  of  these 
kinds,  depending  on  the  right  ascension. 

As  an  additional  test,  the  differences 
of  mean  parallax  were  computed  for  each 
field  separately,  and  compared  with  the 
corresponding  expectations.  The  aver- 
ages, without  regard  to  sign,  for  all  the 
fields,  are  shown  in  table  16. 

The  observed  differences  of  parallax  for  groups  of  stars  differing  in  posi- 
tion on  the  plate  or  in  spectral  type  are  practically  identical  with  those  which 

•No  stars  lying  between  6k  and  iob  were  observed  at  three  or  more  epochs. 


Average. 

Expectation. 

Inner  minus  Outer. 
Bright  minus  Faint. 
White  minus  Red.. 

0^025 
0.028 
0.024 

oTo24 
0.024 
0.026 

DISCUSSION   OF   THE   OBSERVATIONS.  49 

might  be  expected  if  they  had  been  chosen  at  random.  It  follows  that  the 
distortion  of  the  field  and  the  errors  depending  on  the  color  of  the  stars  are 
quite  insensible. 

Bright  and  faint  stars  show  somewhat  greater  differences  of  parallax 
than  groups  chosen  at  random.  There  appears  to  be  some  real  cause  at 
work  here.  Its  average  influence  for  any  given  series,  as  determined  from 
the  difference  of  the  squares  of  the  observed  and  expected  values,  is  oToi4. 
This,  however,  should  hardly  be  called  a  systematic  error,  for  it  appears  to 
vary  quite  at  random  from  one  series  to  another. 

Of  the  31  differences  "Bright  minus  Faint,"  17  are  positive  and  14  are 
negative;  16  are  less  than  the  probable  error  computed  from  their  numerical 
average  and  15  greater.  If  they  are  arranged  in  order  of  right  ascension 
there  are  18  changes  and  13  permanences  of  sign. 

The  source  of  this  error  presumably  lies  in  the  individual  plate.  It 
is  very  probably  "guiding  error" — due  to  imperfect  following — which  can 
not  quite  be  eliminated  by  even  the  best  instrumental  means.  Since  the 
average  weight  of  a  parallax  is  2.98,  an  average  displacement  of  the  bright 
stars,  relative  to  the  faint,  of  0^024,  taking  place  quite  at  random  from  plate 
to  plate,  is  sufficient  to  produce  the  observed  result.  Such  an  error  for  any 
given  star  will  simply  increase  the  accidental  error  of  observation  by  a  small 
amount;  and  the  probable  error,  as  determined  by  comparison  of  different 
plates,  will  include  its  full  effect.  It  need  not,  therefore,  be  further  consid- 
ered, as  it  will  introduce  no  real  systematic  error  into  the  final  parallaxes. 

§6.  Conclusion:  Systematic  Errors  Apparently  Insensible. 

It  may  be  concluded  from  the  preceding  discussion  that  there  are  no 
systematic  errors  affecting  a  star's  observed  parallax,  dependent  either  upon 
its  magnitude,  its  position  in  the  field,  the  character  of  its  own  light,  or  upon 
seasonal  or  instrumental  changes,  of  greater  magnitude  than  a  few  thou- 
sandths of  a  second  of  arc.  In  other  words — 

The  observed  parallaxes  appear  to  be  altogether  free  of  sensible  system- 
atic error. 

This  conclusion  is  all  the  more  satisfactory  because  serious  doubts  have 
been  expressed  concerning  the  possibility  of  obtaining  photographic  positions 
of  high  precision  with  instruments  in  which  a  mirror  forms  part  of  the  optical 
train.  It  is  clear  that  such  fears  may  be  laid  aside — at  least  under  the  cir- 
cumstances of  the  present  work.  It  may  be  observed,  however,  that  these 
are  much  more  favorable  to  accuracy  than  are  the  conditions  in  many  of  the 
most  familiar  instances  in  which  mirrors  are  used. 

Comparison  with  solar  observations,  where  the  mirrors  must  be  exposed 
to  direct  sunlight,  would  be  manifestly  unfair.  As  compared  with  the  ordi- 
nary reflecting  telescope,  the  coude  has  the  advantage  that  its  mirror  is  flat. 
Mere  linear  expansion  or  contraction,  without  deformation,  does  not  affect 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


its  definition  at  all,  while  a  parabolic  mirror,  under  similar  circumstances, 
varies  in  focal  length.  It  is  therefore  not  surprising  that  the  temperature 
effects,  which  are  often  troublesome  in  such  instruments,  are  here  prac- 
tically absent. 

II.  THE  PROPER-MOTIONS. 

§7.  Comparison-stars  with  Sensible  Proper-motion. 

The  real  differences  of  parallax  among  the  comparison-stars  are  too 
small  to  be  detected ;  is  the  same  true  of  the  proper-motions?  The  individual 
proper-motions  of  the  comparison-stars  are  usually  less  than  the  errors  of 
observation,  but  there  are  a  few  stars  which  show  pretty  clear  evidence  of 
motion.  Those  whose  proper-motion  in  x  exceeds  0^20  are  as  follows : 

TABLB  17. 


Bonn  Durch- 

mustcrung. 

Photometric 
magnitude. 

Sp. 

Proper- 
motion  in  x. 

Weight. 

Observed 
parallax. 

Series. 

+43°        55 
-!  J-°      619 
+49°     1956 
+  11°    4776 
+43°    4434 

4-  i°   ,4784 

97* 
10.29 
10.26 
8.10 

9  53 
10  06 

K 

X? 
Gi 
Gt 
A 

+0^35 
-0.38 

-o  37 

+0.2} 
-0.37 
—  O.2I 

i  .  i 
0.9 

39 

1.2 

1  .  I 

-oTo82 

+0.001 

—0.029 
+0.043 
+0.030 
+0.004 

2 

5 
10 

28 
30 

11 

The  above  are  the  proper-motions  referred  to  the  mean  of  the  remaining 
comparison-stars.  The  weights  are  those  found  in  the  least-squares  solutions 
for  the  parallax-stars  of  the  same  series.  As  will  appear  below,  the  probable 
error  of  the  unit  of  weight  is  about  ±oTo57.  Their  mean  observed  parallax 
is— oToos  and  the  average,  without  regard  to  sign,  is  0^031.  If  the  first 
star,  which  is  clearly  affected  to  an  unusual  degree  by  observational  error, 
be  excluded,  the  mean  parallax  is  -foToio.  By  Kapteyn's  formulae  the  mean 
parallax  of  these  stars,  relative  to  the  comparison-stars,  should  be  -f  oToiS. 

§  8.  Average  Proper-Motion  of  the  Rest. 

Excluding  these  six  stars,  the  proper-motions  of  the  remaining  compar- 
ison-stars may  be  investigated  as  follows.  If  the  31  complete  series  are 
arranged  according  to  the  weight  with  which  the  proper-motion  is  deter- 
mined, they  fall  naturally  into  four  groups  constituted  as  follows: 

I.  Series  1-6,  15,  23-26,  30,  31. 
II.  Series  10,  n,  14,  17,  18,  20. 

III.  Series  7-9,  28. 

IV.  Series  12,  13,  16,  19,  21,  22,  27,  29. 

The  first  consists  of  the  fields  photographed  at  three  epochs  only,  the 
second  of  those  observed  at  four  consecutive  epochs;  while  the  remaining 
series  fall  into  two  groups,  in  one  of  which  the  weight  of  y  is  between  3  and  4, 
and  in  the  other  from  7  to  12. 


DISCUSSION   OF  THE   OBSERVATIONS. 


The  average  value  of  the  observed  parallax  and  proper-motion,  without 
regard  to  sign,  for  the  stars  of  these  groups,  together  with  their  mean  weights, 
as  found  in  the  least-square  solutions  for  the  parallax  stars,  are  given  in 
table  1 8. 


TABLE  18. 


Group. 

No  of 
stars. 

Average 
parallax. 

Weight 
of 
parallax. 

Average 
proper- 
motion. 

Weight 
of  P.  M. 

Expecta- 
tion. 

True 
proper- 
motion. 

I 

IOI 

0:034 

3.58 

0:059 

°  93 

0*057 

oToi; 

II 

47 

0.034 

3.12 

0.051 

1.70 

0.046 

O.O22 

III 

29 

0.029 

2.96 

0.031 

3-39 

0.027 

0.016 

IV 

59 

0.032 

3-57 

O.O22 

10.52 

0.019 

O.OI2 

The  observed  proper-motions  decrease  rapidly  with  increasing  weight, 
showing  that  they  are  principally  due  to  accidental  error.  If  they  were 
entirely  due  to  this  cause,  the  mean  proper-motion  and  parallax  for  each 
group  should  be  in  the  inverse  ratio  of  the  square  roots  of  their  weights.  In 
this  way  the  quantities  in  the  column  headed  "Expectation"  are  derived 
from  the  observed  mean  parallaxes.  These  are  uniformly  smaller  than  the 
observed  proper-motions.  The  most  reasonable  explanation  of  the  discrep- 
ancy is  that  the  stars  really  have  small  but  sensible  proper-motions.  If  these 
are  distributed  in  a  random  manner,  the  square  of  the  observed  average 
proper-motion  will  be  equal  to  the  sum  of  the  squares  of  the  average  real 
proper-motion  and  the  accidental  error.  Thus  the  "true"  average  proper- 
motions  given  in  the  last  column  are  determined. 

The  observed  proper-motions,  by  themselves,  can  be  represented  as  the 
results  of  accidental  error  about  as  well  as  the  parallaxes.  The  average  error 
of  a  single  plate  can  be  found  from  each  of  the  tabular  quantities  by  multi- 
plying it  by  the  square  root  of  its  weight.  This  gives  the  results  shown  in 
table  19. 

But  the  two  values  of  the  average 
error,  derived  in  this  way,  are  incon- 
sistent. There  is  some  systematic 
cause  of  discrepancy  at  work,  which 
causes  the  plates  to  agree  better  when 
the  proper-motions  of  the  stars  are 
eliminated  than  when  they  are  merely 
assumed  to  be  negligible;  and  this 
must  almost  certainly  be  real  proper- 
motion. 

If  each  of  the  tabular  values  of  the  real  motion  be  given  a  weight  pro- 
portional to  the  product  of  the  number  of  stars  in  the  group  by  the  weight 
of  an  individual  proper-motion,  the  mean  is  o!oi4.  With  this  value  of  the 
average  real  motion,  and  the  accidental  errors  given  above,  the  observed 


TABLE  19. — Average  error  of  one  plate. 


From  the 
parallaxes. 

From  the 
proper-motions. 

Group  I  

°*°55 

oTo6i 

Group  II   

o  060 

0.066 

Group  III   .... 

O  O4Q 

0.058 

Group  IV  

y 

0.061 

0.072 

Weighted  mean  . 

0.057 

0.064 

DETERMINATIONS  OF  STELLAR  PARALLAX. 


TABLE  20. 


proper-motions  are  represented  as  shown  in  table  20,  the  agreement  being 
very  satisfactory.* 

The  mean  proper-motions  of  the  stars 
of  each  spectral  type  may  be  derived  in 
the  same  fashion.  To  find  the  "expected" 
values  of  the  proper-motion  it  is  here  neces- 


Observed. 

Computed. 

Group  I  .... 
Group  II  ... 
Group  III.  . 
Group  IV... 

0*059 
0.051 
0.031 

O.O22 

0*059 
0.048 
0.031 
0.023 

sary  to  calculate  the  average  value  of  77= 

v P 

(where  p  is  the  weight  of  the  determination) 
for  the  stars  of  each  group  separately.  The  influences  of  accidental  error 
upon  the  observed  parallaxes  and  proper- motions  will  be  in  the  ratio  of 
the  resulting  means. f 

The  results,  including  all  the  comparison-stars  whose  spectra  were  deter- 
mined, are  as  follows: 

TABLE  21. 


Spectrum. 

No.  of 
stars. 

Observed 
parallax. 

Average 
i 

»/? 

Observed 
proper- 
motion. 

Average 
i 

1    P 

Expecta- 
tion. 

True  average 
proper-motion. 

A 

40 

0*031 

0.60 

0:065 

0.86 

0*044 

0*048 

F 

53 

0.033 

0.58 

0.043 

0.75 

0.042 

O.OIO 

G 

65 

0.033 

0.57 

0.046 

0.69 

0.039 

0.024 

K 

58 

0.036 

0.59 

0.047 

0.75 

0.046 

o.on 

The  mean,  weighted  according  to  the  number  of  stars,  is  o?o26  for 
Type  i  and  oToiS  for  Type  n.  In  view  of  the  inevitable  uncertainties  of 
the  method,  no  stress  can  be  laid  on  the  differences  between  these  individual 
values,  but  it  is  noteworthy  that  all  four  types  show  evidence  of  real  proper- 
motion.  The  mean  for  all  the  stars  is  oTo2i5.  This  includes  five  stars 
(listed  above)  whose  average  proper-motion  is  0^34.  Excluding  these,  the 
mean  proper-motion  of  the  remainder  is  oToi4,  in  satisfactory  agreement 
with  the  result  previously  found  with  a  different  grouping. 

Since  the  observed  proper-motions,  upon  which  the  above  calculations 
are  based,  satisfy  three  conditions  for  every  field  (their  sum  for  the  north- 
and-south  and  the  east-and-west  halves  of  the  plate  vanishing)  the  real 
proper-motion  of  the  stars  will,  on  the  average,  be  greater  than  the  tabular 
values  in  the  ratio  i/n  :  '/w-3,  where  n  is  the  number  of  comparison-stars 
on  the  plate.  The  average  value  of  n  is  7.8.  The  average  proper-motion 
of  the  comparison-stars,  excluding  the  six  already  mentioned,  is  therefore 

oToi4X  \  —  or  0^017,  in  x.     Including  the  six  stars,  the  average  proper- 
4.8 


*These  and  many  similar  quantities  have  been  calculated  with  one  more  place  of  decimals  than  appears 
in  the  tables— which  accounts  for  some  apparent  inconsistencies. 

tit  will  not  do  in  this  case  to  calculate  mean  weights  and  then  take  the  square  root,  because  of  the  great 
range  in  the  individual  weights  of  the  proper-motions. 


DISCUSSION   OF   THE  OBSERVATIONS. 


53 


motion  in  x  is  0*025.    Their  average  proper-motion  on  a  great  circle  (if  the 
proper-motions  are  distributed  at  random  on  the  celestial  sphere)   must 

be  -  times  as  great — that  is  0^032., 

7T 

§9.  Proper-Motion  of  the  Parallax-Stars.    Reality  of  the  Observed  Corrections. 

The  corrections  derived  from  the  plates  for  the  catalogued  proper- 
motions  of  the  parallax-stars  may  be  similarly  discussed.  To  render  the 
material  homogeneous,  Bossert's  proper-motions  have  been  used  whenever 
possible.* 

Forming  means  by  groups  as  above  (but  combining  the  two  middle 
groups  on  account  of  the  smaller  number  of  stars  contained  in  them),  the 
results  are: 

TABLE  22. 


Group. 

No.  of 
stars. 

Average  correction 
(without  regard 
to  sign). 

Average, 
weight. 

Average  probable 
error  of  y. 

Average  probable 
error  of  one  plate. 

I        .... 

iS 

0^045 

o  06 

**=oTo4Q 

±0^048 

II  and  III.. 
IV  

13 
n 

0.046 
0.042 

2  33 

IO.7S 

±0.031 

±O  OI2 

±0.043 

±O.O}Q 

It  should  be  noticed  that  the  average  probable  error  of  one  plate  (arith- 
metical mean)  decreases  steadily  with  increasing  weight  of  y,  that  is,  with 
increasing  length  of  time  covered  by  the  observations.  This  may  be  partly 
due  to  chance,  but  it  affords  a  strong  presumption  against  the  existence  of 
any  serious  changes  in  the  instrumental  conditions  with  the  time,  which  would 
show  themselves  by  an  increase.  The  results  for  the  comparison-stars  (see 
p.  51)  confirm  this  conclusion. 

In  this  case,  however,  there  is  no  such  conspicuous  decrease  of  the 
corrections  with  increasing  weight  as  was  apparent  among  the  uncorrected 
proper-motions  of  the  comparison-stars.  In  the  first  group  n  out  of  18 
corrections  are  less  than  their  probable  error;  in  the  second  group,  4  out  of 
13,  and  in  the  third  group  none  at  all.  Of  values  greater  than  three  times 
their  probable  error,  there  are  none  in  the  first  group,  two  in  the  next,  and 
eight  in  the  last.  It  would  therefore  appear  that  many  of  these  corrections 
must  have  a  real  meaning.  They  are  no  doubt  due  in  part  to  errors  in 
the  catalogued  proper-motions  and  in  part  to  proper-motion  of  the  com- 
parison-stars. The  relatively  small  average  for  the  first  group  may  be 
explained  by  the  fact  that  it  contains  all  the  bright  stars,  observed  with 
the  color-screen,  whose  catalogued  proper-motions  are  the  most  accurate. 
This  is  confirmed  by  the  comparison  of  the  proper-motions  used  above  with 
the  very  accurate  ones  of  Professor  Boss's  Preliminary  General  Catalogue 
(which  he  very  kindly  communicated  to  the  writer,  in  advance  of  publica- 

*These  corrections  therefore  differ  from  the  values  of  y  found  in  the  least-square  solutions,  when  the 
proper-motion  used  in  preparing  the  observations  for  the  latter  was  different  from  that  mentioned  above. 
The  orbital  motion  of  >;  Cassiopeiz  has  been  allowed  for. 


54 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


tion,  for  the  18  stars  which  appear  in  the  table).  The  average  correction  to 
the  tabular  values,  regardless  of  sign,  is  oTooS  for  the  five  bright  stars  and 
0^025  for  the  thirteen  others.  The  proper-motions  of  the  remaining  stars, 
not  included  in  the  above  catalogue,  are  presumably  known  with  less  accuracy, 
but  the  assumption  that  their  errors  are  great  enough  to  account  for  the 
whole  of  the  discrepancies  revealed  by  the  plates  is  obviously  violent. 

That  these  are  due  in  many  instances  to  proper-motion  of  the  compari- 
son-stars is  shown  by  discussion  of  the  cases  where  proper-motion  was  evident 
in  one  of  the  latter.  Table  23  gives  the  corrections,  resulting  from  the  plates, 
to  the  catalogued  proper-motion,  before  and  after  the  rejection  of  the  com- 
parison-stars known  to  be  in  motion. 

TABLE.  23. 


Star. 

Correction  to  tabular  proper-motion. 

Probable  error. 

Before  rejection. 

After  rejection. 

Groombridge  34  

—  0^013 
—0.124 
+0.093 
-0.108 
+0.064 
+0.023 
+0.069 
-0.035 
+0.046 
+0.015 
+0.027 

+0^023 
-0.083 
+0.049 
—0.140 
+0.016 
—0.026 
+0.086 
—0.013 

+0.001 

+0.008 
—0.024 

±oToi8 
±0.068 
±0.068 
±0.031 
±0.017 
±0.018 
±0.024 
±0.012 
±0.034 
±0.022 
±0.049 

26  Andromedae*  

p  Persei*   

Groombridge  1646*  

83"  Leonis*  

83*  Leonis*  

Lalande  25372  

Lai.         41402.  . 

Lat.         45755  

Lai.         46650  

Lam.       3*805  

In  seven  cases  out  of  eleven  the  discordance  is  diminished  by  the  rejec- 
tion of  the  moving  comparison-star.  In  two  others  it  is  slightly  increased 
numerically ,  but  not  to  an  amount  unreasonable  in  view  of  its  probable  error. 
In  the  two  remaining  cases  the  discordance  is  increased,  and  is  large  compared 
with  its  probable  error.  For  Lai.  25372  it  may  be  partly  due  to  error  of  tabu- 
lar proper-motion ;  but  this  can  not  be  the  case  for  Groombridge  1646,  whose 
proper-motion,  according  to  Professor  Boss,  is  very  well  determined  and 
shows  no  evidence  of  being  variable.  Proper-motion  of  the  comparison-stars 
could  produce  so  large  a  correction,  without  evidence  of  marked  relative 
motion,  only  if  a  considerable  number  of  them  were  moving  together.  It 
may  be  that  the  difficulty  is,  after  all,  due  to  some  concealed  form  of  error  of 
observation ;  but  the  possibility  remains  that  their  proper-motions  are  really 
variable.  As  both  stars  have  sensible  parallaxes,  the  actual  velocities  which 
need  be  assumed  are  not  great,  and  if  the  orbital  motion  was  of  short  period 
the  amplitude  would  be  small  enough  to  escape  detection  from  the  meridian 
observations  of  the  star.  Whatever  may  be  the  explanation,  the  fact  remains 
that  the  catalogued  proper-motions  of  the  parallax-stars  differ  from  their 
motions  relative  to  the  stars  chosen  for  comparison  by  amounts  which  are 

•The  "tabular  "  proper-motions  for  these  stars  are  those  of  Boss's  Preliminary  General  Catalogue. 


DISCUSSION   OF  THE   OBSERVATIONS. 


55 


often  much  too  great  to  be  disregarded.  Those  solutions  of  the  equations  of 
condition  in  which  these  differences  are  taken  into  account  have  therefore 
been  regarded  throughout  as  definitive. 

III.  THE  "TWO-EPOCH-  PARALLAXES. 

§  10.  Reliability  of  these  Results.    Their  Probable  Errors. 

There  are  six  series  (xxxii  to  xxxvn)  for  which  observations  could  be 
secured  at  but  two  parallactic  epochs  (owing  to  the  accident  to  the  color- 
screen)  and  from  which  the  parallaxes  of  the  principal  stars  were  derived 
with  the  aid  of  the  catalogued  proper-motions.*  The  value  of  the  results 
may  be  tested  by  applying  the  same  process  to  the  stars  of  the  series  which 
were  fully  observed — confining  the  discussion  to  those  for  which  at  least  two 
plates  are  available  at  each  of  the  first  two  epochs  (as  is  the  case  for  all  the 
incomplete  series). 

A  comparison  of  the  results  with  those  of  the  least-square  solutions 
(including  proper-motion  corrections)  is  given  in  the  following  table.  In  this 
pt  —  p,  denotes  the  difference  of  the  mean  parallax-factors  for  the  first  and 
second  epochs  (which  is  positive  when  the  star's  conjunction  with  the  Sun, 
negative  when  its  opposition,  falls  in  the  interval).  P  denotes  the  parallax 
derived  from  the  observations  of  these  two  epochs  and  the  catalogued  proper- 
motion,  and  TT  denotes  the  final  parallax. 

TABLE  24. 


Star. 

Pl-P» 

P 

P-T 

Star. 

Pi-p, 

P 

P-T 

2 

+  1-43 

+oT245 

—  0^005 

30 

-1.03 

+oTno 

+oToij 

3 

+  1-43 

-0.075 

—0.049 

3' 

-1.51 

+0.282 

—0.009 

I 

+  1.76 

+o.  126 

—  O.OIO 

33 

-1.51 

+0.314 

+0.008 

6 

+  1.41 

+0.130 

+0.047 

33 

—  1.21 

+0.087 

+O.OII 

7 

+  1.4' 

—0.007 

—0.014 

34 

-0.93 

—0.004 

+0.007 

ii 

+  1.63 

+0.081 

+0.003 

35 

-0.93 

+0.045 

—0.030 

13 

-1.58 

+0.356 

+O.OIO 

36 

—  1.20 

—  O.OII 

+0.007 

>4 

+  1.40 

+0.111 

—0.052 

37 

+  1.31 

+o  473 

+0.067 

17 

—    -57 

+O.III 

+O.OII 

38 

+  1.3" 

+0.415 

+0.054 

18 

—    -44 

+0.046 

—0.026 

39 

+1.31 

+O.O2I 

+0.056 

19 

—    -44 

+0.040 

—0.014 

40 

+  '•37 

+0.004 

—0.017 

31 

—    -47 

+0.198 

—0.023 

4' 

+  1.30 

+O.J20 

+0.062 

22 

-    .40 

+0.059 

o.ooo 

4' 

+  1-35 

+O.o6o 

+0.023 

23 

-    -4° 

+0.044 

+0.056 

43 

+  '  34 

+0.218 

+0.007 

28 

-    .36 

+0.029 

+0.009 

44 

+  1.34 

—0.007 

+0.015 

29 

-    .36 

+0.082 

+0.02} 

| 

The  mean  value  of  P-  TT  (taking  account  of  sign)  is  -f  oToia  for  the  15  stars 
for  which  p,  —  p3  is  positive,  and  -f  0^003  for  the  16  for  which  it  is  negative, 
or  -foTooS  for  all  together.  The  average  value,  regardless  of  sign,  iso!o27, 
so  there  is  no  evidence  of  systematic  difference  between  the  two  groups. 

Some  difference  between  P  and  TT  should  be  caused  by  the  accidental 
errors  of  observation.  On  the  assumption  (which  is  approximately  true) 
that  an  error  has  the  same  numerical  influence  on  the  observed  parallax,  no 
matter  on  what  plate  it  occurs,  this  effect  may  be  calculated. 

*Pages  8,  35- 


56  DETERMINATIONS  OP  STELLAR  PARALLAX. 

Let  a  be  the  mean  of  m  independent  quantities  of  probable  error  r, 
b  that  of  n  others,  and  c  that  of  all  together.  Then  the  probable  error  of 

(a-b)  will  be  r  J—  -{--,  and  that  of  c,  r  \l— ^— .    But(a-c)  =  — ^—  (a-b), 
*m     n  ^m+n  m+n 

whence  its  probable  error  is  r  \ — ; r,  which  is  \  —  of  that  of  c.  If  then  P 

^m(m-\-n)  *m 

depends  upon  m  plates,  and  TT  upon  these  and  n  additional  ones,  the  prob- 
able error  of  P— v  should  be  \—  of  that  of  TT. 

\m 

The  average  values  of  m  and  n  for  the  stars  of  table  24  are  4.65  and 
2.90;  the  average  probable  error  of  TT  is  ±0^0277;  and  the  corresponding 
average  error,  regardless  of  sign,  is  0^0328. 

The  accidental  errors  of  observation  will  therefore  account  for  a  dis- 
crepancy between  P  and  TT,  of  average  amount  0^0328 X  \— —  or  oTo26 — as 

4-65 

against  0^027  observed.    There  is,  therefore,  no  sensible  systematic  differ- 
ence between  P  and  IT. 

The  parallaxes  derived  from  observations  at  two  epochs  and  the  cata- 
logued proper-motions  are  therefore  entitled  to  confidence.*  They  are  less 
accurate  than  those  derived  from  longer  series,  but  the  difference  is  mainly 
due  to  the  increased  number  of  observations  in  the  latter  and  the  correspond- 
ing diminution  of  accidental  error. 

The  probable  errors  deduced  from  the  residuals  are,  however,  not  those 
of  T,  but  of  (v+by),  where  b  is  a  numerical  coefficient,  whose  values  (given 
in  table  C)  range  in  absolute  magnitude  from  0.37  to  0.17,  and  y  is  the  excess 
of  the  proper-motion,  relative  to  the  comparison-stars,  above  the  catalogued 
proper-motion. 

The  values  of  this  quantity  for  the  five  stars  observed  at  three  epochs 
with  the  color-screen  (using  Boss's  proper-motions)  are: 

/3  Cassiopeiae —  0^053 

T]                     +0.033 

p  Persei +o. 093 

/3       "     -0.041 

y  Virginis  (mean) +o. 094 

The  mean  without  regard  to  sign  is  0^063,  corresponding  to  a  probable 
error  of  =*=  0^053.  This  is  greater  than  the  actual  probable  error  arising  from 
y,  for  it  includes  the  effects  of  accidental  error  of  observation.  If  we  take 
this  as  the  probable  error  of  y,  and  combine  the  probable  errors  of  (ir+by)  and 
by  as  if  they  were  independent,  we  will  certainly  obtain  a  sufficiently  large 
value  for  that  of  IT.  To  avoid  all  possibility  of  understatement,  this  has  been 
done  in  the  final  table  of  parallaxes  for  these  stars. 

'This  might  not  be  the  case  where  there  were  but  two  comparison-stars,  and  no  check  upon  the  proper- 
motions  of  these. 


DISCUSSION   OF  THE   OBSERVATIONS.  57 

IV.    ACCIDENTAL  ERROR  OF  STAR-PLACES 
§11.  Types  of  Error.     Notation. 

It  has  already  been  seen  that  there  is  no  evidence  of  sensible  systematic 
error  in  the  measured  star  positions — at  least  of  such  a  character  as  to  affect 
the  deduced  parallaxes.  Their  accidental  errors,  however,  deserve  study. 

The  error  of  the  x-coordinate  of  a  star  (such  as  is  given  in  Table  C), 
derived  from  the  mean  of  the  measures  of  several  exposures  on  one  plate, 
arises  from  several  sources.  These  are 

1 i )  Error  of  measurement  proper — giving  rise  to  differences  between 
successive  measures  of  the  same  image  and  reseau-lines. 

(2)  Errors  peculiar  to  the  individual  image — which  may  be  due 

to  many  causes — bad  seeing,  unequal  sensitiveness  of  the  plate, 
etc.  These  affect  the  real  position  of  the  center  of  the  image 
or  re"seau-lines,  but  vary  in  a  random  manner  from  one  image 
to  another. 

(3)  Errors  peculiar  to  the  individual  plate,  which  may  arise  from 
guiding  error,  systematic  distortion  of  the  film,  etc.    These 
affect  all  the  images  of  the  same  star  on  this  plate  to  the  same 
extent,  but  for  different  plates  may  be  regarded  as  of  random 
character. 

(4)  Errors  due  to  changes  in  the  instrumental  conditions  from  time 

to  time.  These  would  cause  the  agreement  of  two  plates,  taken 
at  a  long  interval,  to  be  worse  than  that  of  two  taken  a  few 
days  apart,  when  allowance  was  made  for  the  motion  of  the 
stars  in  the  interval.  Only  errors  of  this  type  can  give  rise 
to  systematic  errors  in  the  results  of  observation.  From  the 
evidence  already  obtained,  it  appears  that  they  must  be  very 
small  in  the  case  of  the  present  work. 

(5)  In  addition  to  these,  there  is  the  personal  error  of  bisection, 
differing  systematically  for  stars  and  reseau-lines  and  depend- 
ing also  on  the  appearance  of  the  images.    This  has  been  elim- 
inated from  the  mean  results  for  each  plate  by  measuring  half 
the  images  with  the  plate  in  one  position  and  the  other  half  with 
the  plate  turned  through  180°.    (See  Chapter  I,  §  9,  pp.  15, 16.) 

Before  proceeding  to  determine  the  magnitude  of  these  errors,  it  is  well 
to  fix  a  notation  for  them.  Let  the  average  value,  without  regard  to  sign,  of 
the  measurement-error  (i)  be  m,  that  of  the  image-error  (2)  be  n,  of  the  plate- 
error  (3)  be  p,  and  of  the  error  (4)  due  to  instrumental  changes  be  /.  All 
these  are  presumably  independent;  that  is,  they  may  be  expected  to  show 
little  or  no  correlation,  and  their  combined  effect  may  be  found  upon  the 
principles  of  the  theory  of  errors. 

The  personal  error  of  bisection  (5)  is,  by  hypothesis,  eliminated  by 
reversal  of  the  plate,  and  is  the  same  for  all  the  images  of  the  same  star.*  For 

•Any  error  of  bisection  that  fails  to  satisfy  these  two  conditions  will  be  combined  with  the  image-error 
or  the  measurement-error,  respectively. 


58  DETERMINATIONS  OF  STELLAR  PARALLAX. 

different  stars  it  may  be  different.  Let  B  be  its  mean  value  for  any  plate, 
and  b  the  average  variation  from  star  to  star.  The  first  is  a  constant;  the 
second  may  be  regarded  as  varying  at  random  unless  the  stars  are  especially 
selected. 

It  will  be  convenient  to  express  the  average  value  of  these  errors  (and 
the  observed  quantities  from  which  they  are  derived)  in  linear  measure  on 
the  plate  as  well  as  in  the  corresponding  angular  values.  As  they  are  all 
small,  the  most  convenient  unit  is  the  micron.  In  terms  of  the  quantities 
previously  used  we  have  i  .0/1  =  0.00020  re"seau-intervals — which  corresponds 
to  0*0352. 

§  12.  Error  of  Measurement. 

To  find  the  value  of  the  measurement-error  m,  it  is  necessary  to  take 
plates  on  which  each  image  was  measured  in  both  positions.  The  difference 
between  the  mean  of  the  measures  of  two  images  of  each  star  in  the  "direct" 
position  of  the  plate  and  of  the  other  two  "reversed,"  and  that  of  the 
measures  of  the  same  images  in  opposite  positions,  will  be  due  wholly  to  the 
measurement-error  (since  the  image-error  is  independent  of  the  direction  of 
measurement  and  the  bisection-error  is  eliminated  in  the  mean). 

The  average  effect  of  this  error  upon  each  mean  will  be  }4m,  and  upon 

their  difference,  — T=. 

From  the  measures  of  319  stars  on  31  plates,  the  average  difference 
between  such  means  is  0.86/1.  The  corresponding  value  of  m  is 

1.22/1,  or  0^043 

The  average  difference  between  the  mean  result  when  each  star-image 
is  measured  only  once,  in  the  manner  described  above,  and  that  of  measuring 
each  star  in  both  positions  is  only  0.43  /i,  or  0^015.  This  is  practically  neg- 
ligible in  comparison  with  the  other  errors  of  the  plates,  and  on  this  account 
the  shorter  method  of  measurement  was  used  for  the  rest  of  the  work. 

The  personal  error  of  bisection  B  can  be  found  from  the  differences  of 
the  "direct"  and  "reversed"  measures  of  each  image — the  errors  of  measure- 
ment, which  influence  individual  results,  practically  disappearing  from  the 
mean.  The  average  value  of  this  difference  for  the  3 1  plates  is  +4.85/1.  Since 
B  changes  sign  upon  reversal  its  value  is  half  this:  that  is,  2.43/1  or  oToSs. 
This  is  one  twenty-first  of  a  division  of  the  scale  with  which  the  measures 
were  made.  The  individual  values  for  different  plates  show,  however,  a 
much  larger  range  of  variation  than  the  errors  of  measurement  will  account 
for  (the  average  difference,  regardless  of  sign,  between  an  individual  value 
and  the  general  mean  being  22  per  cent  of  the  latter).  It  therefore  appears 
that  the  personal  error  of  setting  is  subject  to  considerable  fluctuations- 
depending  not  only  upon  the  appearance  of  the  images,  but  on  the  physical 
condition  of  the  measurer,  and  perhaps  upon  such  factors  as  the  illumination 
of  the  field  as  well. 


DISCUSSION   OF  THE   OBSERVATIONS.  59 

§13.  Error  Peculiar  to  the  Individual  Image. 

The  image-error  n  may  be  found  from  the  "average  residual"  for  the 
plates.  For  the  3  1  plates  already  mentioned  the  mean  of  the  two  measures 
of  each  image  is  free  from  bisection-error,  but  is  affected  by  measurement 
and  image  error.  The  resultant  of  these  may  be  called  the  internal  error 
(since  it  is  derived  from  the  "internal  agreement"  of  the  measures  on  the 
plate)  and  its  average  value  denoted  by  i.  Then  since  each  image  was  meas- 
ured twice,  i*  =  ri*  +  t^m*  . 

The  "average  residual"  for  these  plates  was  obtained  as  follows:*  The 
differences  between  the  measured  coordinates  of  each  star  for  a  given  expos- 
ure and  their  mean  for  all  the  exposures  were  taken,  and  compared  graphi- 
cally with  an  expression  of  the  form  by+c.  The  average  of  the  residuals, 
without  regard  to  sign,  is  the  tabular  quantity. 

Since,  on  the  average,  there  are  ten  stars  on  each  plate,  this  process 
involves  the  representation  of  40  observed  coordinates  by  18  unknowns 
derived  from  them  (10  mean  coordinates  of  the  stars  and  8  constants  b  and  c). 

The  "average  residual"  must  therefore  be  multiplied  by  \l  —  42  —  to  find  the 

40  —  1  8 

true  value  of  i. 

For  these  3  1  plates  the  mean  '  'average  residual"  is  1.58(1.  Hence  i  =  2  .  1  4/1, 
or  0*075,  and  with  the  value  of  m  found  above  n  =  1.96/1  =  0^069.  If  each 
image  had  been  measured  but  once,  the  average  internal  error  (excluding 


bisection-error)  would  have  been  J/m'+n2,  or  2.3o/x  =  oT 

On  the  remaining  plates  each  image  was  measured  but  once.  It  is 
therefore  impossible  to  find  separate  values  of  the  measurement  and  bisection 
errors  for  them. 

The  internal  error  can,  however,  be  found  —  but  not  directly  from  the 
"average  residual,"  which  in  this  case  includes  also  the  effect  of  the  variable 
part  of  the  error  of  bisection  (which,  though  eliminated  from  the  mean  of 
the  measures  of  the  four  exposures,  influences  their  individual  differences 
from  this  mean). 

The  differences  between  the  measures  for  the  two  exposures  measured 
in  the  same  position  of  the  plate  are,  however,  free  from  bisection  error.  As 
there  is  no  evidence  of  any  real  difference  of  orientation  between  the  succes- 
sive exposures  on  a  plate,*  these  differences  should  be  constant,  and  their 
deviation  from  their  mean  will  measure  the  internal  error  of  the  plate. 

These  differences  were  calculated  for  the  suitable  plates  of  every  fifth 
series  (*'.  e.,  those  with  four  exposures,  not  already  discussed),  numbering  35 
in  all.  Their  average  deviation  from  the  means  for  each  set  is  3.17/1-  The 
average  error  of  such  a  difference  should  be  t'i/2,  and  the  average  deviation 

from  the  mean  of  n  such  differences  should  be  i  -y-     —  .     The  average  value 
of  n  is  in  this  case  9.3.     Hence,  for  these  plates,  i=  2.38/1. 

•Chapter  III.  p.  3'- 


60  DETERMINATIONS  OF  STELLAR  PARALLAX. 

The  "average  residual"  for  these  plates  is  2.15/4.    If  the  internal  error  is 


computed  from  this  the  result  will  not  be  i,  but  J'i'+ft1,  where  b  is  the 
variable  part  of  the  error  of  bisection.  In  this  case  36  observed  coordinates 
have  been  represented  by  13  unknowns  (9  mean  coordinates  of  the  stars  and 
4  means  of  the  differences  for  each  exposure).  Therefore 


For  the  remaining  plates  with  four  exposures  and  full  weight,  121  in 
number,  the  mean  average  residual  is  2.  33/1,  whence,  as  above,  i/ 


If  b  has  the  same  value  for  these  plates  as  for  the  preceding  ones,  which 
form  a  large  and  apparently  typical  sample  of  the  whole,  then  for  these  last 
plates  *  =  2.64)u. 

The  mean  of  the  three  values  of  i,  with  weights  proportional  to  the  num- 
ber of  plates  on  which  each  is  based,  is  i  =  2.  53/1  =  0*089. 

If  the  measurement-error  for  all  the  plates  is  the  same  for  those  upon 
which  it  could  be  determined,  then,  since  i*  = 


§14.  Error  Peculiar  to  the  Individual  Plate. 

The  error  peculiar  to  the  plate,  p,  must  be  found  by  comparison  of  pairs 
of  plates  taken  within  a  few  days  of  one  another,  for  which  the  real  motions 
of  the  stars  and  the  possible  instrumental  changes  are  presumably  negligible. 

The  average  error  k  of  a  mean  coordinate  derived  from  four  exposures 
on  such  a  plate  will  be  given  by  the  equation  k'  =  p'-\-^i*. 

The  average  discordance  between  the  mean  results  for  the  two  plates 
would  be  k*/2  if  the  plate  constants  used  in  comparing  them  were  exactly 
known.  But  since  there  are  three  of  these  constants,  the  average  discordance 

for  the  n  comparison-stars  used  in  determining  them  will  be  k\j?-^- 


n 

7\ 

since 


For  the  parallax  stars  the  average  discordance  will  be  k\2^n'~1',  si 

n 

for  them  the  uncertainty  of  the  correction  to  reduce  to  standard,  whose 
weight  is  approximately  n,  is  added  to  the  (independent)  error  of  the  meas- 
ured coordinates. 

Since  on  the  average  plate  there  are  7.9  comparison  and  1.4  parallax 
stars,  the  average  discordance  will  be  1.115^  for  the  former  and  1.501^  for 
the  latter,  which,  considering  their  numbers,  makes  the  general  mean  1. 173^. 

Taking  only  plates  of  full  weight,  86  pairs  are  available.  The  average 
discordance  of  the  coordinates  of  such  a  pair,  after  reduction  to  standard,  is 

i  .99//;  whence 

k  =1.70^  =  0".  060 

With  the  value  of  i  already  found 


DISCUSSION  OF  THE  OBSERVATIONS.  6  1 

The  relative  displacement  of  bright  and  faint  stars,  varying  at  random 
from  plate  to  plate,  whose  average  amount  was  estimated  in  §5*  as  0*024  (in 
addition  to  the  other  errors  which  vary  without  reference  to  the  magnitude 
of  the  stars)  is  of  the  type  of  error  here  considered  and  will  be  included  in  the 
value  of  p  just  obtained.  These  groups  of  bright  and  faint  stars  contain  on 
the  average  about  four  members.  Between  the  means  for  any  two  such 
groups,  chosen  at  random,  there  should  be  a  difference,  owing  to  the  plate- 

error,  of  -—  or  0^028.    It  therefore  appears  that  a  considerable  part  of  the 

plate-error  depends  on  the  magnitude  of  the  stars  —  which  confirms  the 
opinion  that  guiding  error  is  an  important  factor  in  its  production  —  prob- 
ably exceeding  all  other  causes  combined.  This  being  the  case,  it  is  apparent 
that  the  plate-error  would  not  be  wholly  eliminated  by  making  all  the 
exposures  for  a  single  series  at  successive  epochs  on  one  plate,  according 
to  Professor  Kapteyn's  plan.  This  would  indeed  get  rid  of  such  errors  as 
arise  from  distortion  of  the  film  or  of  the  reseau-lines  ;  but  the  guiding-error, 
which  is  not  influenced  at  all  by  the  making  of  previous  or  subsequent  expos- 
ures on  the  same  plate,  or  by  the  method  of  measurement,  would  have  the 
same  effect  as  ever. 

§15.  Error  Due  to  Instrumental  or  Seasonal  Changes. 

There  remains  the  error  t,  due  to  instrumental  changes.  The  value  of 
this  can  be  found  by  comparing  the  average  error  of  a  determination  of  paral- 
lax, calculated  from  the  agreement  of  plates  taken  at  a  few  days  interval, 
with  that  actually  observed. 

k 
The  theoretical  expression  for  the  former  is  —  =,  where  p  is  the  weight 

IP 

of  the  determination  of  parallax.     The  average  value  of  -j=  for  the  31 

. 
series  is  0.60  1  ;  whence  the  average  error  of  one  parallax  if  no  instrumental 

errors  are  present,  should  be  0*036. 

The  actual  value  may  readily  be  found  in  the  case  of  the  parallax-stars. 
The  arithmetical  mean  of  the  probable  errors  of  their  parallaxes  is  0*028. 
The  corresponding  average  error,  without  regard  to  sign  (obtained  by  divid- 
ing this  by  0.845)  is  0*033.  But,  by  the  reasoning  of  the  last  section,  this 

must  be  multiplied  by  \-~,  or  0.942,  to  allow  for  the  effect  of  the  errors 

*  «+i 

of  the  plate  constants.    The  final  value  for  these  44  stars  is  therefore  0*031. 

The  average  value,  without  regard  to  sign,  of  the  observed  parallaxes  of 

the  242  comparison  stars  is  0*033.    To  allow  for  the  errors  of  the  plate- 


constants,  this  must  be  multiplied  by     --  ,  or  1.270,  giving  0*042. 

If  the  observed  parallaxes  were  wholly  due  to  errors  of  observation  the 
average  error  of  one  parallax  for  all  the  stars  would  be  0*040,  and  the  part 


*Page  49. 


62  DETERMINATIONS  OF  STELLAR  PARALLAX. 

of  this  due  to  instrumental  changes  would  be  oToiy.  But  this  is  undoubt- 
edly too  great.  The  observed  parallaxes  of  the  comparison-stars  are  influ- 
enced, in  addition  to  the  errors  of  observation,  by  (i)  the  errors  of  the 
approximate  method  used  in  deriving  them,  and  (2)  the  real  differences  in 
parallax  between  the  stars.  The  first  of  these  quantities  can  be  determined 
by  comparison  of  the  least-squares  and  approximate  solutions  for  the  parallax- 
stars.  The  average  difference,  without  regard  to  sign,  between  the  values 
of  the  parallax  obtained  in  the  two  ways  is  0^005.  The  actual  differences  in 
parallax  among  the  comparison-stars  are  more  difficult  to  estimate.  An 
attempt  may,  however,  be  made  in  two  ways: 

(a)  The  comparison-stars  are  selected  by  magnitude  alone,  without 
respect  to  their  proper-motion.    The  only  group  of  stars  of  known  parallax 
which  satisfies  the  same  condition  is  that  of  the  brightest  stars,  all  of  which 
have  been  observed.    From  the  table  given  in  the  Annals  of  the  Cape  Obser- 
vatory (vol.  vin,  part  II,  p.  1423)  it  follows  that  the  mean  parallax  of  22  such 
stars  is  oTioS,  while  the  average  value,  regardless  of  sign,  of  the  differences 
of  the  individual  parallaxes  from  the  mean  is  0^117,  and  the  mean-square 
value  of  these  differences  is  oTi74.     This  surprising  state  of  affairs  results 
from  the  fact  that  a  few  of  the  stars — notably  a  Centauri — have  very  large 
parallaxes  whose  differences  from  the   mean  are  much  greater  than  the 
mean  itself,  while  several  others  have  very  small  parallaxes,  so  that  their 
residuals  are  negative  and  almost  numerically  equal  to  the  mean. 

The  errors  of  observation  for  these  parallaxes  are  not  great  enough  to 
have  any  serious  influence  on  the  above  values. 

According  to  Kapteyn's  formulae,  the  mean  parallax  of  the  comparison- 
stars  is  0^0057.  By  analogy  with  the  bright  stars,  we  might  therefore  expect 
the  average  difference,  regardless  of  sign,  of  one  parallax  from  the  mean  to 
be  oToo6,  and  the  mean-square  difference  0^009. 

(b)  The  only  direct  determination  of  such  differences  of  parallax  among 
the  faint  stars  appears  to  be  that  given  by  Kapteyn  (Groningen  Pub.  No.  20, 
p.  27).    After  a  thorough  elimination  of  the  accidental  errors  of  observation 
and  the  errors  depending  on  magnitude,  he  finds  evidence  of  real  differences 
of  parallax  (among  3600  stars  in  eight  areas  in  different  parts  of  the  sky) 
whose  average  amount  corresponds  to  a  probable  error  of  ±0^017.     The 
photographs  upon  which  this  determination  was   based  were,  however, 
exposed  at  widely  different  hour-angles  for  the  morning  and  evening  obser- 
vations, so  that  systematic  differences  of  the  observed  parallax,  arising  from 
atmospheric  dispersion*  and  depending  on  the  spectral  type  of  the  stars,  may 
affect  the  results. 

As  no  spectroscopic  data  are  available  for  these  stars  (most  of  which 
are  too  faint  to  appear  in  the  Bonn  Durchmusterung),  such  differences 
can  not  be  distinguished  from  the  real  parallaxes,  and  the  quantity  just 

•Kapteyn  calls  attention  to  this  and  emphasizes  the  importance  of  confining  the  exposures  to  a  fixed 
hour-angle  in  future  work,  a  policy  which  he  was  the  first  to  propose. 


DISCUSSION   OF  THE  OBSERVATIONS. 


quoted  is  the  resultant  of  both.  It  appears,  however,  from  accompanying 
data,  that  the  quantity  given  by  the  observations  is  approximately  TT  — o.  55/3 
(where  TT  is  the  parallax  and  5/3  the  difference  of  the  refraction  constant,  rela- 
tive to  the  mean  of  all  the  stars  on  the  plate).  The  coefficients  of  5/3  for  the 
individual  plates  vary  somewhat,  without  departing  far  from  this  average 
value.  As  5/3  may  in  some  cases  be  as  great  as  oTio,  it  is  clear  that  its  vari- 
ations may  account  for  the  greater  part  of  the  real  differences  among  the 
observed  quantities.  In  the  absence  of  data  as  to  its  average  amount,  it  is 
impossible  to  derive  from  these  data  the  real  average  difference  of  parallax 
among  these  stars. 

It  is  necessary,  therefore,  to  fall  back  on  the  estimate  (a).  Taking  the 
mean-square  value  for  the  variability  of  the  real  parallax  (0*009)  and  the 
value  0*005  f°r  the  average  error  due  to  the  approximate  method  of  solution, 
the  average  influence  of  all  errors  of  observation  on  an  observed  parallax 
of  a  comparison-star  becomes  0*041,  and  that  for  all  the  stars  0*039,  so  that 
the  effect  of  the  instrumental  error  is  0*016. 

The  corresponding  average  error  /  of  one  star-coordinate  may  be  found 
by  dividing  this  by  0.60,  whence 

/  =  o*75M  =  0*026 

If  Kapteyn's  value  for  the  real  differences  of  parallax  should  be  adopted 
without  correction  for  the  error  depending  on  spectral  type,  the  average 

value  of  this  difference,  without  regard  to  sign,  would  be  -    ~-  or  0*020. 

0.054 

The  influence  of  errors  of  observation  upon  the  parallax  of  a  comparison- 
star  is  then  reduced  to  0*037,  and  for  all  the  stars  together  to  0*036,  leaving 
nothing  to  be  explained  by  instrumental  changes ;  but,  for  the  reasons  already 
stated,  it  seems  that  this  tempting  procedure  is  not  warranted. 

§  16.  Summary  of  Results.     Best  Number  of  Exposures  per  Plate. 

The  average  errors  already  found  may  be  converted  into  probable  errors 
by  multiplying  by  0.8543.  The  results  are  as  shown  in  table  25.  These  are 
average  values  for  all  stars  and  all  plates.  The  error  of  bisection,  which  is 
eliminated  by  reversal  during  measurement,  is 

0.0024=1=0.0011  mm.  (0*085 ± 0*037) 
the  probable  error  measuring  its  variation  from  star  to  star. 

The  average  probable  error  of  one  mean  star-coordinate,  derived  from 
plates  with  different  numbers  of  exposures,  is  given  in  table  26. 


TABLE  25. 


TABLE  26. 


Seat  of  error. 

Probable  error. 

In  mm. 

In  arc. 

Measurement  (tn)  

±0.0010 
±0.0019 
±0.0010 
±0.0006 

±0^036 
±0.066 
±0.034 
±0.022 

Image  (n)        .             

Plate  (p) 

Instrumental  change  (0  

Exposures. 

Probable  error. 

2    

±0^067  or  ±0.0019  mm. 

4      

±0.055  or  ±0.0016 

6        .   ... 

±o  051  or  ±0.0015 

DETERMINATIONS  OF   STELLAR  PARALLAX. 


Certain  important  conclusions  can  be  drawn  from  table  25.  The  errors 
m,  n,  p,  t  are  very  nearly  in  the  ratio  3:6:3:2.  Their  respective  contributions 
to  the  total  error  of  the  mean  result  from  a  plate  with  four  exposures  are 
*4>  36,  37,  and  16  per  cent.  That  is,  the  measurement-error  contributes 
but  one-seventh  of  the  whole.  To  have  measured  all  the  star-images  in 
both  positions  of  the  plate  would  have  decreased  the  probable  error  of  the 
mean  by  only  3  per  cent — a  ridiculously  insignificant  return  for  the  labor 
involved.  The  policy  of  measurement  adopted  for  the  major  part  of  the 
work  is  thus  conclusively  confirmed. 

The  number  of  exposures  per  plate  has  an  important  bearing  on  the 
efficiency  of  the  work.  To  save  measurement  it  must  be  even.  The  probable 
error  of  the  resulting  mean  coordinates  may  be  divided  into  two  parts,  one 
independent  of  the  number  of  exposures  and  the  other  decreasing  with  it. 
If  we  take  the  former  as  our  unit  (that  is,  ^ p'-\-f  or  0^040),  the  probable  error 

of  an  equation  derived  from  a  plate  with  n  exposures  will  be  0^040  \  i  +— , 


n 


and  its  weight  proportional  to 


211 


To  determine  the  efficiency,  it  is 


TABLE  27. — Work  necessary  to  secure 
equal  accuracy  with  different 
numbers  oj  exposures. 


necessary  to  estimate  the  relative  labor  which  must  be  expended  in  obtain- 
ing such  equations. 

The  work  of  taking  the  plates  and  measuring  them  is  very  nearly  pro- 
portional to  the  number  of  exposures.  That  of  development,  etc.,  and  that 
of  reduction  are  the  same  for  all.  The  latter  is  certainly  less  than  the  former. 
Under  the  conditions  of  the  present  work  it  may  be  estimated  to  be  between 
one-half  and  three-fourths  as  great  (for  a  plate  of  four  exposures).* 

The  total  labor  cost  of  a  plate  of  n  exposures  will  therefore  be  propor- 
tional to  some  quantity  between  «+2  and  n+$.  On  these  two  hypotheses 
the  values  of  the  quotient  of  work  divided  by  weight  are  as  shown  in  table  27. 

It  is  clear  that  there  is  a  considerable  loss  in 
making  more  than  six  exposures  on  one  plate. 
For  smaller  numbers  the  efficiency  is  nearly  con- 
stant. In  practice  it  is  found  that  to  make  six 
exposures  on  each  plate  seriously  diminishes  the 
number  of  fields  that  can  be  observed  each  even- 
ing, while  to  make  but  two  exposures  (on  twice  as 
many  plates)  is  also  rather  inconvenient.  More- 
over, when  there  are  but  two  images  of  each  star 
on  the  plate,  it  is  impossible  to  tell  which  is 
wrong,  in  case  of  trouble.  Thus  the  policy  of 
making  four  exposures  on  each  plate  is  also  justified  by  the  results. 

The  errors  due  to  instrumental  changes  are  almost  surprisingly  small, 
when  it  is  considered  that  they  include  changes  in  the  reseau,  as  well  as  in 

•This  estimate,  and  the  results  deduced  therefrom,  might  be  materially  changed  in  a  case  where  the 
services  of  assistants  were  available  for  measurement  and  reduction. 


n 

Work  proportional  to  — 

n+2 

»+3 

2 
I 

8 

II.  0 
11.2 

12  .7 
14.4 

13.8 
13.1 

I4   2 
I5.8 

DISCUSSION  OF  THE  OBSERVATIONS.  65 

the  telescope  and  its  mirror,  through  more  than  three  years.  They  con- 
tribute less  than  one-sixth  of  the  whole  error  of  an  average  plate  (with  four 
exposures).  About  three-fourths  of  the  error  of  such  a  plate  is  inherent  in 
the  individual  plate  and  images.  This  must  be  the  combined  effect  of  many 
causes — bad  seeing,  imperfect  guiding,  unequal  sensitiveness  of  the  plate, 
distortion  of  the  film,  etc. — whose  relative  importance  can  not  now  be  esti- 
mated. But  it  would  be  very  desirable  to  test  different  kinds  of  plates,  with 
the  same  instrumental  conditions,  in  the  hope  of  reducing  it. 


V.    PROBABLE  ERRORS  FOR  THE  PARALLAX.STARS. 

§17.  Loss  of  Accuracy  for  Close  Double  Stars. 

The  mean-square  value  of  the  probable  error  of  unit  weight  resulting 
from  the  least-square  solutions  for  the  44  parallax-stars  is  ±0^048.  This 
is  less  than  the  average  value  just  found  for  all  the  stars  on  the  plate — which 
is  not  surprising,  since  the  parallax-stars  are  favorably  placed  at  the  center 
of  the  plate  and  were  measured  with  special  care.  The  results  for  different 
classes  of  stars  are,  however,  of  very  different  degrees  of  precision. 

There  are  five  pairs  of  stars*  on  the  list,  whose  images  on  the  plate 
lie  much  closer  to  one  another,  in  the  direction  of  measurement,  than  the 
ordinary  separation  of  the  successive  exposures  (i7?6).  The  mean-square 
probable  error  for  these  ten  stars  is  ±0^0596,  while  for  the  remaining  34 
stars  it  is  ±0^0437. 

It  therefore  appears  that  the  presence  of  a  comparison-star,  of  com- 
parable magnitude,  within  about  10"  (or  0.03  mm.  on  the  plate)  seriously 
diminishes  the  accuracy  of  the  measures.  This  might  be  expected,  owing 
to  mutual  disturbance  of  the  images,  which  under  ordinary  conditions  are 
from  3  to  5  seconds  of  arc  in  diameter.  This  difficulty  is  due  to  the  duplicity 
of  the  stars,  and  not  to  the  photographic  method  of  observation.  When 
the  stars  are  close,  and  of  comparable  brightness,  observations  with  the 
heliometer  (and  doubtless  with  other  instruments  as  well)  are  diminished 
in  accuracy  in  the  same  way  (as  was  found  by  Gill  in  the  case  of  a  Crucis).f 

It  is  clear  that  only  the  isolated  stars  afford  a  fair  test  of  the  accuracy 
of  the  photographs.  Three  of  these,  however  (Nos.  3,  39,  and  44),  are  at 
some  distance  from  the  center  of  the  fields  on  which  they  appear  (being 
observed  because  they  happened  to  lie  on  plates  taken  primarily  for  some 
other  star).  The  mean-square  probable  error  for  these  is  =*=  0^059,  and  for 
the  remaining  31  stars  is  ±0^042.  This  is  again  what  might  be  expected, 
for  the  accuracy  of  the  measures,  and  also  of  the  corrections  necessary  to 
reduce  them  to  standard,  diminishes  toward  the  edge  of  the  field. 

•Nos.  8-9,  15-16,  18-19,  31-32.  34-35.  fCape  Annals,  vol.  vm,  part  n,  p.  SSB. 


66 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


TABLE  28. 


§18.  Dependence  of  Accuracy  upon  Photographic  Magnitude,   and  Position 
on  the  Plate. 

The  probable  errors  for  the  remaining  stars 
(including  all  those  observed  under  normal  condi- 
tions) are  given  below,  arranged  in  order  of  their 
photographicmagnitudes.  The  latterwere  obtained 
by  adding  to  the  visual  photometric  magnitudes 
the  corrections  shown  in  table  28  (given  by  King, 
Harvard  Annals  59,  v,  page  152). 

For  the  bright  stars  observed  with  the  color- 
screen,  the  effective  photographic  magnitude  was 
assumed  to  be  5.5  magnitudes  fainter  than  the 

visual  magnitude  (no  account  being  taken  of  the  spectral  type,  as  it  is  well 
known  that  photographs  taken  through  a  yellow  screen  give  results  which 
in  this  respect  agree  closely  with  the  visual  magnitudes).  The  effective 
magnitudes  so  obtained  are  inclosed  in  parentheses  in  table.  The  last 
column  gives  the  average  duration  of  exposure  for  each  group. 

TABLE  29. 


Spectrum. 

Correction. 

0 

-o?4 

A 

o.o 

F8 

+0.6 

G 

+0.7 

G5 

+0.9 

K 

+  1.2 

K5 

+  1.4 

M 

+  1.7 

Star 
No. 

Photographic 
magnitude. 

Probable  error 
of  one  observation 
of  unit  weight. 

Mean-square 
value  of 
groups. 

Average 
exposure. 

37 

7-0 

-.oTosi         ) 

' 

m 

12 

7-  ' 

43 

17 

7.1 

54 

±0.048 

3-7 

It 

7-5 

34 

26 

7-5 

61 

40 

76 

±0.023 

7-6 

38 

I 

(7-6) 
7-7 

36 
48 

±0.035 

39 

29 

7-8 

27          J 

1 

(79) 

±0.014 

4> 

8.3 

37 

28 

8.6 
9.0 

31 

09 

±0.029 

40 

4 

(91) 

40 

10 

9  i 

±0.011 

6 

(9~  3) 

64 

3 

94 

>9 

±0.035 

43 

36 

94 

34 

30 

95 

31 

21 

97 

k 
XcQ  _  O)3 

35 

9.8 

41 

33 

99* 

80 

- 

14 
5 

24 

10.1 
IO.2 
10.3 

30 
70 
31 

5-° 

43 

10.5 

±0.023 

22 

10.6 

78 

^l 

10.7* 

26 

±0.047 

4.6 

33 

10.8 

48 

30 

11.  of 

39 

•Direct  estimate  from  plates.     fOr  excluding  No.  33,  ±0^042.     {Estimated  (spectrum  not  determined). 


DISCUSSION   OF  THE  OBSERVATIONS. 


The  individual  values  vary  through  a  wide  range.  This  is  not  surprising, 
for  the  number  of  observations  for  a  single  star  (from  five  to  eleven)  is  really 
not  sufficient  to  permit  a  reliable  determination  of  the  individual  probable 
errors  (especially  when  three  unknowns  have  to  be  derived  from  the  observed 
data).  There  is  little  doubt  that  the  very  small  probable  errors  found  for  a 
few  stars  are  evidence,  not  mainly  of  the  exceptional  accuracy  of  the  obser- 
vations, but  very  largely  of  the  fortuitous  coincidence  of  errors  of  nearly  equal 
magnitude  in  the  two  or  three  observations  of  each  parallactic  epoch;  and 
it  is  equally  likely  that  in  some  other  cases  the  probable  errors  in  the  table 
are  unduly  increased  by  the  opposite  accident. 

The  meansof  groups  of  five, however, are  based  on  a  total  of  from  34  to  41 
observations  (that  is,  at  least  19  more  than  the  whole  number  of  unknowns 
derived  from  them)  and  should  therefore  be  reliable  indications  of  the  mean 
error  of  observation  for  stars  of  the  corresponding  magnitudes. 

These  show  a  distinctly  systematic  variation  with  the  magnitude,  the 
only  exception  being  the  last  group  but  one.  The  mean  for  this  is  raised  by 
the  presence  of  one  very  bad  star  (No.  33).  The  original  observations  show 
that  the  abnormally  large  probable  error  for  this  star  is  almost  entirely  due 
to  the  extreme  discordance  of  a  single  plate,  on  which  its  images  were  recorded 
at  the  time  of  measurement  as  excessively  faint.  It  is  questionable  whether 
this  plate  should  have  been  measured  at  all.  In  any  case,  it  does  not  rep- 
resent the  normal  probable  error  for  stars  of  this  magnitude.  A  second  mean 
has  therefore  been  formed  for  this  group,  excluding  this  star.  The  legitimacy 
of  this  process  is  confirmed  by  the  fact  that  if  the  under-exposed  plate  were 
rejected  the  probable  error  derived  for  this  star  from  the  remaining  plates 
would  have  been  close  to  the  mean  for  the  rest  of  the  group.  The  means, 
after  this  correction,  can  be  closely  represented  by  the  formula 

r*  =  io+4(m-9)*  (i) 

where  r  is  the  probable  error,  expressed  in  hundredths  of  a  second  of  arc,  and 
nt  the  photographic  magnitude — as  is  shown  by  table  30. 

The  average  duration  of  exposure  for 
the  different  groups  is  so  nearly  equal  that 
the  extreme  magnitudes  need  be  changed  by 
little  more  than  one-tenth  of  a  unit  to  reduce 
the  results  for  all  the  groups  to  the  average 
exposure  of  4^3 .  The  corrections  to  the  effec- 
tive photographic  brightness  of  the  stars  on 
different  plates,  depending  on  the  clearness 
of  the  air,  etc.,  would  doubtless  be  much 
larger  than  this.  The  above  formula  may 
therefore  be  applied  with  sufficient  approximation  to  all  the  plates. 

For  the  stars  not  at  the  center  of  the  field,  an  additional  term  may  be 
introduced  into  the  expression  (i)  proportional  to  the  square  of  the  distance 
p  from  the  center  of  gravity  of  the  comparison-stars.  The  material  is  not 


TABLE  30. 


Photographic 
magnitude. 

Observed. 

Computed. 

7-2 

0^048 

oTo47 

7.6 

.035 

.041 

8.6 

.029 

.032 

94 

•035 

.032 

IO.2 

.042 

.039 

10.8 

.047 

.047 

68 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


TABLE  31. 


sufficient  to  give  a  reliable  coefficient  for  this  term;  but  with  the  approxi- 
mate expression 

r'  =  io+4  (m-9)'+o.3p3  (2) 

(where  p  is  expressed  in  reseau  intervals)  the  observed  data  are  represented 
in  table  31. 

From  the  formula  (14),  page  25,  setting  r=  ± 0^055,  n  =  -j.8,  P  =  Q  =  5, 
it  appears  that  the  increasing  uncertainty  of  the  correction  to  standard  for 
points  remote  from  the  center  would  alone 
produce  a  term  -f  o.isp*  in  (2).  The  remain- 
der may  be  interpreted  as  showing  a  decrease 
in  the  accuracy  of  the  measures  of  stars  at 
a  distance  from  the  center,  which  may  help 
to  account  for  the  inferior  accuracy  of  the 
measures  of  the  comparison-stars. 

If  the  term  depending  on  the  photographic  magnitude  is  assumed  to 
be  the  same  for  the  close  double  stars  as  for  the  others,  the  probable  errors 
of  observation  may  be  represented  by  the  formula  (see  table  32) 

r*  =  30-r-4(w-9)>  (3) 

TABLB  32. 


Star. 

m 

p 

r  comp. 

r  obs. 

3 

6.0 

4 

±0^072 

±0^069 

39 

9.0 

9 

0.057 

0.055 

44 

9.0 

7 

0.049 

0.050 

Star. 

Photographic 
magnitude. 

Observed  r. 

Mean-square 
value. 

Formula  (3). 

15 

7-5 

±oTos9    ) 

34 

7-8 

34 

*o*°57 

±oTo58 

35 

8.0 

67     f 

16 

8-7 

6>    J 

8 

9.0 

±0.058    ) 

18 
19 

& 

I 

±0.058 

±0.055 

9 

9-$ 

61     J 

3i 

3* 

II.  O 

II.O? 

±0.056    \ 
78    / 

±0.068 

±0.068 

This  satisfactorily  explains  the  unusually  large  probable  error  for  the 
last  pair.  The  observations  of  a  component  of  such  a  double  have,  under  the 
best  conditions,  only  one-third  the  weight  of  those  upon  an  isolated  star  of 
the  same  magnitude. 

For  the  eight  stars  (Nos.  45  to  52)  observed  at  two  epochs  only,  the 
mean-square  probable  error  of  unit  weight,  resulting  from  the  observations, 
is  =to?o53.  That  computed  as  above,  with  the  aid  of  the  formulae  (i)  and 
(3),  remembering  that  Nos.  46  and  47  form  a  close  pair,  is  ±o?O4i. 

In  this  group  there  is  again  one  star  (No.  50)  showing  a  very  large  prob- 
able error  ( ±0^092)  due  to  discordance  of  very  faint  images  on  underexposed 
plates.  If  this  is  excluded  from  the  mean,  we  have  for  the  other  stars  an  ob- 
served mean-square  probable  error  of  ±  0^044,  as  against  ±oTo42  computed. 
It  therefore  appears  that,  except  for  the  one  case  aforesaid,  the  observations 
of  the  interrupted  series  are  closely  comparable  in  accuracy  with  the  rest. 


DISCUSSION   OF   THE   OBSERVATIONS. 


69 


§  19.  Importance  of  Correct  Exposure. 

The  results  for  the  isolated  stars  show  the  great  importance  of  correct 
exposure.  The  range  of  brightness  within  which  measures  of  the  greatest 
precision  can  be  made  on  a  given  plate  is  decidedly  limited,  not  exceeding 
two  magnitudes.  A  departure  of  one  magnitude  on  either  side  of  the  most 
favorable  brightness  decreases  the  weight  of  an  observation  to  seven-tenths, 
and  one  of  two  magnitudes  to  less  than  one-third,  of  its  maximum  value. 
Part  of  this  error  is  due  to  the  character  of  the  images  of  bright  or  faint  stars— 
the  former  being  too  large,  and  often  too  diffuse  at  the  edge,  for  accurate 
measurement,  and  the  latter  too  ill-defined,  often  without  any  definite  center 
to  set  on.  But  it  is  probable  that  some  of  the  other  sources  of  error  are  also 
considerably  increased — e.g.,  guiding  error  for  bright  stars,  and  that  arising 
from  unequal  sensitiveness  of  the  plate  for  faint  ones. 

Over-exposure  or  under-exposure,  sufficient  to  impair  seriously  the 
accuracy  of  the  measures,  can  often  be  recognized  on  inspection  of  the  star- 
images.  With  longer  series  of  plates,  it  would  probably  be  advisable  to 
reject,  a  priori,  any  on  which  the  images  of  the  parallax-star  were  defective 
in  this  respect.  In  the  present  case  this  would  be  too  drastic;  but  such 
plates  were  given  diminished  weight. 

It  would  have  been  well  if  the  exposures  for  the  faint  stars  had  been 
longer.  Those  for  the  bright  stars  could  not  well  have  been  much  curtailed, 
owing  to  the  necessity  of  getting  good  images  of  the  comparison-stars;  but  a 
color-screen  of  small  absorption  (two  or  three  magnitudes)  would  make  it 
possible  to  get  good  images  of  both. 

By  formula  (i)  the  normal  probable  error,  for  an  isolated  star  at  the 
center  of  the  field,  of  the  mean  coordinate  derived  from  one  plate  with  four 
exposures  of  4?  3  each  is  as  shown  by  table  33. 

As  it  is  not  certain  to  what  extent  the  deviations 
from  these  values  for  the  individual  stars  are  due 
to  chance,  and  to  what  extent  they  represent  real 
variations  in  the  accuracy  of  observation,  two  sets  of 
probable  errors  are  given  in  the  final  table  of  observed 
parallaxes ;  the  first  being  that  resulting  from  the  indi- 
vidual least-squares  solution,  while  the  second  is  that 
derived  from  the  weight  of  the  determination  of  the 
parallax  and  the  formula  (i),  (2),  or  (3)  for  the  prob- 
able error  of  one  observation,  according  as  the  star  is:  (i)  an  isolated  star 
at  the  center  of  the  field;  (2)  a  similar  star  in  another  part  of  the  field; 
(3)  a  close  double  star.  The  mean-square  value  of  the  probable  error 
given  by  formula  (i)  for  stars  within  one  magnitude  of  the  most  favorable 
brightness  may  well  be  taken  as  a  measure  of  the  average  accuracy  of 
observation  attainable  in  the  present  research.  This  is  readily  found  to  be 
±0^0356,  which  corresponds  to  ±0.00101  mm.  on  the  plates. 


TABLE  33. 


Photographic 
magnitude. 

Probable 
error. 

7.0    11.  o 

±oTo5i 

75     "0-5 

.044 

8.0    10.0 
8-5      95 

.037 
.033 

9.0      9.0 

.032 

7O  DETERMINATIONS  OF   STELLAR   PARALLAX. 

§20.  Results  for  Stars  close  together  only  Partially  Independent.    Explanation. 

The  measures  of  double  stars  are  of  much  less  precision.  It  has  already 
been  shown  that  special  sources  of  error  exist  in  close  pairs.  In  addition  to 
this,  there  is  reason  to  fear  that  when  we  have  two  stars  of  comparable  magni- 
tude within  a  few  minutes  of  arc  in  the  heavens  (or  a  few  millimeters  on  the 
plates)  the  effects  of  the  instrumental  error  (t)  and  of  the  plate-error  (/>)— 
or  at  least  of  the  guiding  error  which  forms  the  greater  part  of  this — will  be 
practically  identical  for  the  two  stars,  as  will  also  be  any  influence  of  errors 
of  measurement  of  the  comparison-stars,  so  that  the  parallaxes  derived  for 
the  two  from  the  same  plates  will  not  really  be  independent  determinations- 
Twelve  such  pairs  appear  in  the  list  of  the  stars  observed  for  parallax.  In 
every  case  the  two  stars  have  a  considerable  common  proper-motion,  and 
it  is  therefore  practically  certain  that  their  parallaxes  are  sensibly  identical. 
The  difference  of  the  observed  values  will  thus  afford  a  measure  of  those 
errors  of  observation  which  are  not  common  to  the  measures  of  the  two  stars. 

The  mean-square  value  of  this  difference  for  the  12  pairs  is  0^043,  to 
which  corresponds  the  probable  error  =*=oTo29  for  the  difference  of  the  two 
observed  parallaxes  and  ±oTo2i  for  each  singly,  if  their  error  were  inde- 
pendent. But  the  mean-square  probable  error  of  a  determination  of  parallax, 
relative  to  the  comparison-stars,  for  one  of  these  same  stars  is  ^0*035.  It 
follows  that  those  errors  of  observation  which  differ  for  the  two  stars  of 
the  pair  account  on  the  average  for  only  36  per  cent  of  the  whole,  leaving  64 
per  cent  as  the  contribution  of  errors  common  to  the  two  stars  on  the 
same  plate. 

This  is  of  the  order  of  magnitude  which  was  to  be  expected,  for  the  data 
of  p.  63  show  that  for  plates  with  four  images  of  a  star,  each  measured  with 
double  the  usual  number  of  settings  (as  was  the  case  for  the  parallax-stars), 
the  errors  p  and  /  account  for  56  per  cent  of  the  whole  error  of  the  measured 
coordinates  of  this  star.  The  error  of  the  correction,  computed  from  the 
measures  of  the  comparison-stars,  and  necessary  to  reduce  the  measures  to 
standard,  increases  the  square  of  the  error  of  the  reduced  coordinate  by 
about  12  per  cent.  Hence  61  per  cent  of  the  latter  is  due  to  known  sources 
of  error  certainly  or  presumably  common  to  the  two  stars  of  a  pair.  It  is 
therefore  unnecessary  to  go  farther  for  an  explanation  of  the  observed  facts. 

It  follows  that  the  mean  of  the  observed  parallaxes  of  such  a  pair  of 
stars  has  by  no  means  twice  the  weight  of  an  individual  determination. 
According  to  the  data  just  obtained  it  has  1.22  times  that  of  one  of  the  com- 
ponents. 

It  would  therefore  appear  at  first  sight  that  the  photographic  method 
is  in  this  respect  at  some  disadvantage.  But  it  should  be  remembered  that 
the  measurement  of  the  second  component  of  a  pair  on  the  plates  involves 
very  little  additional  work.  The  same  comparison-stars  and  plate-constants 
serve  for  both,  and  the  whole  increment  of  labor  after  the  plates  are  taken 


DISCUSSION   OF  THE   OBSERVATIONS. 


(if  there  are  seven  or  eight  comparison-stars)  is  hardly  over  10  per  cent;  so 
that  the  mean  parallax  of  a  pair  of  stars  (provided  they  are  not  too  close) 
can  be  determined  with  slightly  greater  accuracy,  in  proportion  to  the  work 
expended  on  it,  than  that  of  an  isolated  star. 

§21.  Comparison  of  the  Average  Precision  Attained  by  Different  Observers. 

It  is  of  interest  to  compare  the  accuracy  of  these  results  with  that  of 
other  modern  methods  of  observation.  The  probable  errors  found  by  other 
photographic  or  micrometric  observers  are  directly  comparable  with  these; 
those  of  heliometer  observations  must  be  halved  (since  in  this  case  any  dis- 
placement of  the  central  star  produces  a  change  twice  as  great  in  the  meas- 
ured difference  of  distances. 

Table  34  shows  the  accuracy  with  which  the  relative  position  of  a  star  has 
been  determined  by  some  of  the  best  modern  observers  and  methods,  and 
the  relative  weights  of  an  average  observation.  The  unit  weight  corresponds 
to  a  probable  error  of  0^05,  and  the  means  are  in  all  cases  "mean-square." 

TABLE  34. 


Method 
or 
instrument. 

Observer. 

No.  of 
series. 

Mean 
probable 
error. 

Wt. 

References. 

Heliometer: 

[Chase  (earlier)  . 

83 

=*=O  OO4 

O  } 

Yale 

\  Chase  (later) 

At: 

=tn  O7I 

O   5 

Yale  Transactions,  Vol    n   part 

[  Smith  

tt 

±0  O72 

' 

o  || 

i.  Dage  i  OH. 

Leipzig  

B.  Peter  

14 

•fco  048 

1.  1 

Abhd.  d.  Kgl.  Sachs.  Gesellsch. 

[Gill     . 

IO 

±o  o}6 

2   O 

derWissenschaften  xxn,  xxiv, 
xxvn,  xxx. 

Cape  (y-inch)  .... 

JFinlay  

3 

=*=o  047 

1  .  1 

Cape  Annals  vin,  part  n,  page 

[de  Sitter. 

4 

±o  057 

o  8 

n6s. 

Micrometer: 
Yerkes  4O-inch  . 

Barnard  .  . 

I 

=*=o  060 

O.7 

Monthly  Notices,  LXVIII,  p.  637 

Photography: 
Upsala 

±o  oso 

1  .O 

(computed    from    data    there 
given). 

Astronomische  Nachrichten  3999. 

Bonn  (i  i  -inch) 

(Kustner  1 

2 

=*=O  O14 

2   2 

Mean-square  probable  error  for 

Yerkes  (4O-inch) 

\KapteynJ  '  ' 
Schlesinger  

±O.OJO 

T.   8 

stars   all  over   the   plate  for 
mean  of   three   exposures    at 
one  epoch.     Computed  from 
data  given  in  Pub.  Ast.  Lab. 
Groningen  2),  pages  55-56,  etc. 
Probable  error  of  single  exposures 

Present  Work: 
Close  doubles 

IO 

=•=0.060 

O.7 

physical  Journal  20,  page  116. 

All  other  stars  .  .  . 

34 

±0.044 

1.3 

Stars  with  correct 
exposure,    For- 
mula (i)  

=•=0.036 

2.0 

The  average  accuracy  of  the  present  work  has  been  considerably  dimin- 
ished by  the  inclusion  of  stars  observed  under  unfavorable  circumstances. 


72  DETERMINATIONS  OF  STELLAR  PARALLAX. 

For  isolated  stars  (which  alone  furnish  a  fair  basis  of  comparison)  the  aver- 
age of  all  its  results,  good  and  bad,  is  exceeded  in  precision  by  but  a  single 
observer  using  other  than  photographic  methods.  The  observations  made 
under  suitable  conditions  of  exposure  (which  could  always  be  realized  in 
future  work)  are  seriously  surpassed  only  by  the  plates  taken  with  the  great 
Yerkes  telescope,  whose  focal  length  gives  it  a  great  advantage  over  smaller 
instruments. 

In  view  of  its  high  precision  and  of  the  comprehensive  evidence  ob- 
tained of  its  freedom  from  perceptible  systematic  errors  (at  least  in  the  case 
of  the  present  work),  the  photographic  method  may  fairly  claim  to  be  estab- 
lished in  the  front  rank  as  a  means  of  determining  stellar  parallax. 


CHAPTER  V. 

RESULTS  OF  OBSERVATION.* 

I.  RESULTS  OF  THE  PRESENT  WORK. 

§  i .  Description  of  Table  A . 

The  observed  parallaxes  of  those  stars  which  have  been  the  special 
objects  of  investigation  are  collected  in  table  A,  pp.  76-77. 

The  first  seven  columns  of  this  table,  giving  the  current  number  of  the 
star,  its  designation,  place  for  1900,  photometric  magnitude  and  spectral  type 
(as  determined  at  Harvard),*  and  its  proper-motion,  are  practically  identical 
with  the  corresponding  columns  in  table  C  and  are  repeated  here  for  con- 
venience. The  last  five  columns,  also  taken  from  table  C,  show  in  what 
series  of  plates  each  star  appears,  the  number  of  comparison-stars,  of  plates 
in  the  series,  of  exposures  measured,  and  the  average  exposure  time. 

The  eighth  column  contains  the  observed  parallax,  relative  to  the  com- 
parison-stars, as  derived  from  the  least-squares  solutions  in  which  proper- 
motion  terms  were  included.  The  tenth  column  gives  the  probable  errors 
derived  directly  from  these  solutions,  and  the  eleventh  those  derived  from 
the  weights  of  the  determinations  of  parallax  and  the  general  expressions 
for  the  probable  error  of  one  observation,  derived  in  Chapter  IV,  §19  (page 
69),  and  depending  on  the  photographic  magnitude. 

The  number  of  plates  (*'.  e.,  of  equations  of  condition)  from  which  the 
observed  parallaxes  are  derived  is  so  small  that  the  probable  errors  derived 
directly  from  the  residuals  are  subject  to  very  considerable  uncertainty.  On 
the  other  hand,  it  is  quite  possible  that  the  assumption  that  all  observations 
of  stars  of  the  same  photographic  magnitude  are  of  equal  accuracy  (on  which 
the  formulae  for  probable  error  are  based)  goes  too  far.  Both  sets  of  values 
are  therefore  given.  By  choosing  the  greater,  one  can  pretty  surely  be  on 
the  safe  side;  but  this  is  hardly  fair  to  the  observations.  In  the  remainder 
of  this  work,  the  mean-square  average  of  the  two  has  been  adopted  except 
in  six  cases — Nos.  5,  6,  18  (y  measures),  33,  50,  and  52 — in  which  the  excess 
of  the  directly  derived  probable  error  is  due  to  the  large  discordance  of  one 
or  two  observations  in  each  case.  As  in  such  instances  it  is  clear  that  some 


*For  two  stars,   the  Harvard  data  are    lacking,  or 
uncertain.      Their   spectral   types   have   been    estimated 
from  the  difference  between  the  known  visual  magnitude  and 
the  photographic  magnitude  determined  by  comparison  with 
the  other  stars  on  the  plates  (using  for  these  the  visual 
magnitudes  corrected  for  spectral  type  according  to  King's 
determination).    The  visual  and  photographic  magnitudes 
of  these  stars  and  the  concluded  spectral  types  are  shown 
in   the   accompanying   table.     The   results  are  given  in 
parentheses  in  Table  A. 

No. 

20 
41 

Visual 
magni- 
tude. 

Photo- 
graphic 
magni- 
tude. 

P-V 

Spectrum. 

9.93 
9  41 

II.  0 

10  8* 

+1.0 
+1.4* 

GS 

K5  (Harvard  K?) 

73 

74 


DETERMINATIONS  OF   STELLAR  PARALLAX. 


of  the  errors  of  observation  must  be  unusually  great,  the  larger  values  given 
by  the  direct  reckoning  are  retained.  These  adopted  values  are  given  in  the 
ninth  column  immediately  after  the  parallaxes. 

In  the  case  of  pairs  of  stars  with  common  proper-motion,  the  simple 
mean  of  the  observed  parallaxes  is  taken  as  the  definitive  value  for  both. 
The  differences  in  weight  corresponding  to  the  directly  derived  probable 
errors  are  for  the  most  part  illusory,  and  even  those  resulting  from  the  for- 
mulae just  mentioned  are  too  great,  for,  as  shown  in  Chapter  IV,  §20  (p.  70), 
the  greater  part  of  the  error  of  observation  is  common  to  the  two  stars. 

The  probable  errors  of  these  means  are  therefore  considerably  greater 
than  they  would  be  if  the  two  determinations  were  strictly  independent. 
In  accordance  with  the  discussion  just  quoted,  they  have  been  taken  as  0.90 
times  the  mean-square  average  of  the  probable  errors  for  the  two  components. 

For  the  stars  numbered  from  45  to  52,  which  were  observed  at  two 
epochs  only,  the  probable  errors  derived  as  above  have  been  increased  to 
the  extent  demanded  by  the  assumption  that  the  assumed  values  of  their 
proper-motion  (relative  to  the  comparison-stars),  which  were  used  in  deriving 
the  parallaxes,  have  a  probable  error  of  ±0^053.  (See  page  56.) 

When  the  difference  between  the  proper-motions  previously  assumed 
for  these  stars  and  those  given  in  Boss's  Catalogue  was  great  enough  to  have 
a  sensible  influence  on  the  deduced  parallax,  the  necessary  corrections  were 
made. 

In  the  two  cases  where  the  parallax  was  derived  independently  from 
measures  of  the  x  and  y  coordinates,  the  resulting  values  have  been  combined 
with  weights  according  to  their  observed  probable  errors. 

The  notes  concerning  table  A  explain  themselves.  They  deal  mainly 
with  the  numerous  double  stars. 

NOTES  TO  TABLE  A. 

The  data  concerning  double  stars  are  taken  from  Burnham's  "General  Catalogue  of 
Double  Stars,"*  unless  otherwise  noted;  the  remainder  are  principally  from  Lewis's  memoir 
on  the  Struve  stars  (Mem.  R.  A.  S.  i,vi).  Magnitudes  given  to  two  decimal  places  are 
derived  from  photometric  measures  made  at  Harvard,  some  of  which  may  be  found  in  the 
Harvard  Annals,  vol.  LXIV,  No.  vi.  The  relative  motion  given  in  the  case  of  double  stars 
is  that  of  the  fainter  component  referred  to  the  brighter.  The  relative  masses  in  certain 
binary  systems  are  taken  from  the  notes  to  Boss's  "Preliminary  General  Catalogue." 

(1)  Bu.  24;  204°,  22?6  (1900);  Mags.  2.42,  13.7;  Optical  pair;  Companion  not  shown  on 

plates;  Observed  with  color-screen. 

(2)  Triple;  Mags.  7.73,  10.5,  11.5;  (the  last  two  rough  estimates). 


A  B 

A  C 

Date. 

Observer. 

Remarks. 

»?0      tO*.A 

1866.23 

Auwers.f 

B  is  a  physical  and  C  an  optical  compan- 

56. a    38.9 
56.3    38.7 

iu?6    34^7 
118.0    30.3 

1904.98 
1906.78 

Plate  391. 
Russell,  in 

ion.    The  proper-motion  of  A  accounts 
for  the  change  in  the  latter.    Not  in  Bu. 

•Referred  to  hereafter  as  Bu. 


.  Ablh.  Berlin  Akad.  der  Wissensch.  1867,  p.  23. 


RESULTS  OP  OBSERVATION.  75 

(3)  Bu  131;  O2  5;  Mags.  6.04,  10;  241°,  6".i;  Fixed;  Measured  because  it  was  on  the 

plates  of  Series  n;  Companion  shown  on  plates,  but  not  measurable;  Proper- 
motion  given  ten  times  too  great  in  A .  G.  Bonn. 

(4)  Bu.  426;  2  60;  Mags.  3.67,  7.41;  227°,  5?6  (1904);  Binary;  Period  long  and  uncertain. 

Annual  motion  of  B  relative  to  A  (1904)  from  Lewis's  diagram,  — 0^19  in  x,  +oTi9 
in  y.  Mass  of  B  0.76  that  of  A  (Boss);  Observed  with  color-screen;  Companion 
not  shown. 

(5)  Variable  (Mira) ;  Mag.  1.7  to  9.6;  Period  331.6  days;  Radial  velocity  constant,  +63  km. 

(Campbell  and  Stebbins,  Astrophysical  Journal,  vol.  18,  page  341).  The  distant 
(optical)  companion  is  comparison-star  5;  79°,  116^9  (1904);  Bu.  1209. 

(6)  Variable;    Mag.  3.4  to  4.2;    Irregular;  Obs.  with  color-screen. 

(7)  Variable  (Algol)  Mag.  2.1 103.2;  Period  2d  2oh  48m  55';  Spectroscopic  triple;  the  close 

pair  having  the  period  of  the  light  variation  and  also  revolving  about  the  center  of 
mass  of  the  system  in  a  nearly  circular  orbit  with  radius  not  less  than  89,000,000  km. 
and  period  1.899  years  (Curtiss,  Science  N.  S.,  vol.  28,  page  848).  Observations 
extending  over  at  least  two  years  are  necessary  to  separate  the  effects  of  this  orbital 
motion  and  the  annual  parallax.  The  present  series,  completed  before  this  fact  was 
known,  covers  only  one  year,  and  its  results  must  therefore  be  regarded  as  provisional. 
Three  faint  and  distant  companions;  Bu.  1565.  Observed  with  color-screen. 

(8,9)  Bu.  1854;  2  443;  48°,  876  (1897);  Combined  mag.  7.83.  Physical  pair;  Relative 
motion  — oToi4  in  *,  -f-o?oo3  in  y  (Lewis). 

(n)  Lalande  9012. 

(15,  16)  Bu.  5779;  21540;  150°,  29?2  (1892);  Combined  mag.  6.04;  Physical  pair;  Relative 
motion  per  year  — o7oo6  in  *,  +oToio  in  y  (Lewis). 

(18,  19)  Bu.  6243;  2  1670;  328°,  5*9  (1903);  Combined  mag.  2.91;  Binary;  Period  182 
years,  a  =3^90;  Masses  equal  (Boss).  Relative  motion  of  following  star  +0^040  in 
x,  —0^003  in  y  (Lewis).  Observed  with  color-screen. 

(20)  Proper-motion  from  A.  G.  Berlin  A.  Comparison  of  this  catalogue  with  the  plates 
gives  the  proper-motion  relative  to  four  comparison-stars  as  +0^030,  — 1^85,  which 
is  much  nearer  the  value  deduced  from  the  plates  themselves. 

(22,23)  Bu.  6869;  75°,  45?2  (1904);  Relative  motion  very  small;  Proper-motion  from 
Porter  Pub.  Cincinnati  Observatory,  vol.  15,  page  100. 

(26,27)  Bu.  7162;  21919;  10°,  24! i  (1905);  Combined  mag.  6.41 ;  Physical  pair;  Relative 
motion  —  oTooi  in  x,  —  oToo6  in  y  (Lewis). 

(29)  Bu.  7332;  02298;  185°,  i?2  (1903);  Binary;  Period  56  years;  a  =oT88;  Not  sepa- 
rated on  the  plates.  Difference  of  magnitude  between  the  components  0.3  (Bu.); 
which  makes  the  individual  magnitudes  7.4  and  7.7. 

(28,  29)  Have  common  proper-motion  and  are  relatively  fixed  in  328°,  121^9. 

(31,32)  Bu.  8798;  22398;  150°,  17?!  (1900).  Combined  mag.  8.87;  Binary;  Period 
long;  Relative  motion  —  oTo5  in  x,  —o'.oi  in  y  (from  diagram  in  Bu.). 

(34.35)  Cygni  6B.;  Bu.  9137;  22486;  217°,  9^2  (1905);  Combined  mag.  5.97;  Physical 
pair;  Relative  motion  +0^022  in  x,  +oToo2  in  y  (Lewis). 

(36)  Fifth-type  star  with  hydrogen  atmosphere  5*  in  diameter.  No  data  regarding 
proper-motion. 

(37.38)  Bu.  10732;  22758;  127°,  22?5  (1904);  Combined  mag.  5.12;  Binary;  Period  very 
long;  Relative  motion  oToo  in  x,  —0*20  in  y  (Lewis);  Masses  nearly  equal  (Boss). 

(39)  Suspected  by  Kapteyn  to  have  a  parallax  of  about  oTi;  Proper-motion  from  com- 
parison of  the  plates  with  A.  G.  Lund,  using  9  comparison-stars. 

(41)  Bu.  11671;  Kriiger  60;  120°,  3^3  (1905);  Combined  mag.  9.43;  Difference  of  magni- 
tude 1.3,  according  to  Barnard's  estimates;  which  makes  the  individual  magnitudes 
9.7  and  n.o;  Binary;  Period  probably  less  than  100  years;  Distant  optical  com- 
panion in  59?6,  40?2  (1905),  of  mag.  10.23,  used  by  Barnard  as  a  comparison-star 
for  parallax;  Proper-motion  variable,  owing  to  orbital  motion.  Masses  of  the  com- 
ponents comparable.  (Barnard,  M.  N.,  vol.  68,  page  643.)  Tabular  proper-motion 
from  Barnard,  A.  J.,  vol.  23,  page  171.  Only  the  principal  star  measurable  on  the 
plates;  close  companion  invisible. 


76 


DETERMINATIONS  OP  STELLAR  PARALLAX. 


TABLB  A. — OBSERVBD  PARALLAXES. 


NTo 

I 

2 

3 

4 
5 
6 

7 
8 
9 

10 
ii 

13 
13 

14 

:i 
>7 

18 

'9 
18 

'9 
20 

91 
32 

33 
24 

35 

26 

37 

28 

29 

3° 

3' 
32 

33 

34 
35 

36 

12 

39 
4» 
41 

42 
43 
44 

Designation. 

Right 
ascen- 
sion 
1900.0 

Decli- 
nation 
1900.0 

Magni- 
tude. 

y 

• 

& 

Proper- 
motion. 

Paral- 
lax. 

Prob- 
able 
error. 

P.  E. 
from 
obser- 
vations. 

P.  E. 
from 
ormula 

Scries. 

Compari- 
son-stars. 

Plates. 

(ft 

B 

I 

o. 

X 

W 

(u-5 
MM 

£  G 
v£ 

< 

0  Cassiopeiae      .  .  . 

0-3-8 
o  13.7 
o  13.3 

043.0 

3   14.3 

2   58.8 

3     1-7 
}  3  40.2 

+  58°36' 
+43  27 
+43  15 
+  57  "7 
-  326 

+3837 
+40  34 

+4"    9 

2.42 

7  73 
6.04 
3.64 
1.7109.6 
3.4104.2 

2.1  to  3.2 

/    8.35 
I    8.89 

F5 
Ma 
A 
F8 
Md 
Mb 
B8 

}° 

0^56 
2.80 
0.03 
1.24 
0.24 
0.17 

O.OI 

..38 

+0'032 
+0  250< 
-0.026 
+0.187- 
+0.136 
+0  083 
+0.007 
—0.029 

+O.020 

-0.004 
-0.011 
+0.078 
+0.049 

+0.346* 
+0-335 
+0.344 

+0  163 

+0.048 
+0.057 
+0  052 

+0.100 

+0.054 
+0.070 
+0.068 
+0.072 
+0  063 

+0.105 
+0  221 

+0.059 

—  O.OI2 

+0.024 

+0.045 
+0.014 
+0.030 

-0.039 

-0.077 

-0  058 

+O.O2O 
+0.059 

+0.040 
+0.095 

+0.291 
1+0.306 
+0.298 

+0.076 

—  O.OI  1 

+0.075 
+0  032 

-0  018 

+0.406 
+0.361 
+0.384 

-0  035 
+0.021 
+0.258 

+0  037 
+0  211 
-0  022 

±0'019 
±0  016 
±0.042 
±0.019 
±0.035 
±0  040 
±0  027 

±0.033 
±o  034 
±0  030 

±0.014 
±0  019 
±0.023 

±0.015 
±0.032 
±0.014 

±0.018 

±0.034 
±0.033 
±0.031 

±0  029 

±0.032 
±0.028 
±0.157 
±0.071 
±0  026 

±0  024 
±0.020 

±0.036 
±0.027 
±0.029 

±0.022 
±0.023 
±0.021 

±0.051 
±0.038 
±0  041 

±0.013 
±0.019 
±0.015 

±0.016 

±0.042 
±0.049 
±0  041 

±0.065 

±0.049 
±o  063 
±0.050 

±0  022 
±0.024 

±0.021 

±0  021 
±0  029 
±0.023 
±0.019 

±0  021 
±0  023 
±0  033 

±o'ocx) 

0.011 

0.041 

O.O2I 

0.035 
0.040 
0.025 

0.033 
0.035 
0.031 

0.006 
0.013 

0.021 

0.015 
O.O3I 
O.013 

0.015 

0.033 
0.035 
O.O3I 

O.O3O 

0.034 
0.027 
0.157 
0.074 
O.O26 

0.020 
O.OI9 

0.044 
O.O27 
0.033 

O.OI9 
0.025 
O.O2I 

0.058 
0.036 
0.045 

O.OO5 
O.OI4 
0.010 

O.OI2 

0.038 
O.O52 
O.O4I 

0.065 

0.034 
0.067 
0.045 

0.022 

O.O24 
O.O23 
O.O2I 

O.O29 
O.O15 
O.OI3 

O.O2I 
O.OI5 
0.033 

±oTo26 
o  020 
0.043 
0.017 

O.O2O 
O.O2O 

O.O28 

j 

O.O32 
O.O32 
O.O29 

0.019 
O.O23 
0.025 

O.OI5 
0.032 
O.OI4 

0.020 

0.035 
0.031 
O.O3I 

0.028 

0.030 
0.030 
0.067 
0.067 
O.O26 

O.O27 
O.O2O 

0.025 
O.027 
0.024 

O.O2J 
0.021 
0.021 

O.O42 
O.O4O 
0.037 

0.017 
O.O2I 
0.0l8 

O.OI9 

0.045 
0.045 
O.O4I 

O.O29 

O.o6l 
0.059 
0.055 

0.022 

O.024 
0.019 
O.O20 

0.029 
0.028 
0.023 

O.O20 
0.028 
0.022 

}    " 

III 
IV 
V 
VI 

}     VII 

VIII 
IX 
X 

XI 
XIo 

XII 
}    XIII 

XIV 

}      XV 

I    XVa 

XVI 
XVII 

|  XVIII 
1    XIX 
}     XX 

\    XXI 

XXII 
\    XXIII 

XXIV 
\    XXV 

XXVI 
XXVII 

XXVIII 
XXIX 

XXX 

|  XXXI 

9 
9 

8 
9 
9 
8 

6 

8 
8 
8 

}> 
6 
5 

7 
6 

7 
9 

7 
7 
8 
8 

7 
8 

8 
7 

6 
10 

8 
9 
8 

10 

5 
6 

7 
7 
7 

7 
6 

6 
6 

7 
8 

9 
7 

8 
8 

8 
8 

7 
6 

5 

8 

u 
6 

6 

7 

6 

}" 

10 

6 

8 

7 
6 

2O 
34 

35 
31 

23 
24 

24 

23 
21 
29 

3' 

35 

28 

3' 
3" 

32 
30 

27 
26 
3O 

30 

41 
23 

24 

38 

24 

46 

42 
23 
32 

27 
24 

3»3 

3.8 

41 
52 
50 
4-3 
4-5 

42 
42 
35 
4.8 

4.7 
39 

4  • 

4.8 

4.8 
4.6 

52 
5-8 
4.0 

45 

4.8 
45 

4-7 
4.6 

4-8 

2.7 

2.7 
3-8 
39 

3-4 
37 

Groombridge  34  — 
26  Andromedae.  .  .  . 
17  Cassiopeiae   . 

u  Ceti  

p  Persei  

ft  Persei  

Lalande6888  
Lalande  6889    .   ... 

Mean  

Lalande  744} 

3  56-5 
4  44-4 
10  21.9 

10  57.9 

+35    3 
+45  41 
+4921 
+3638 

8.34 
6.83 
6.50 
7-4* 

G2 

G 

F8 
K 

2.20 

0.68 
0.90 
4.78 

Groombridge  884.  .  . 
Groombridge  1646.. 
Lalande  21  185  

From  measures  of  y\ 
Mean          

Lalande  21258 

II    0.5 

jll  31.7 

+44    2 
+  333 

8.63 

/    6.36 
\    7.51 

K5 

K 

K 

4.40 

}°-74 

85  *  Leonis         

8  ^  *  Leonis  

Groombridge  1830.. 

II  47.2 

}ia  36.6 

+3826 
-  o  54 

6.46 

/    3.65 
I    3-68 

G 

)F 

7.04 
0.56 

•y*  Virginis  

From  measures  of  y 
From  measures  of  y 
Mean        

13  40.3 
13  40.7 

JI42I.I 

+  18  20 
+  1526 

+24    6 

9.98 
8.30 

(    9  f 
\    9-64 

(Gj) 
K5 
K 

1.89 
3.33 

},38 

Lalande  25372  

Berlin  B  $072  
Berlin  B  5073   

A   Oe    14318   

15    4-7 
"5    47 

-"5  59 
-15  54 

9.09 
8.86 

K 
G5 

}3-75 

A.  Oe.  14320  

Mean             

Lalande  27742  
Lalande  27741 

}«5    8.3 

+  19  39 

(    6i3 
1    7-63 

G 
A? 

jo.  68 

Mean  

W  B    i5h  716  

•5  33.4 
>5  33.5 

+40  10 
+40    8 

IS 

K 

G5 

}o.48 

W.  B.  15*730  

Mean            

W  B.  17*  322  

17  20.8 
1841.7 

+  2  14 
+  5929 

7.82 

/    9  33 

\    10.01 

Ma 

K5 
K5 

1.36 

H 

>Pos  Med  2164    .. 

Mean           

Lam.  18180  

18  53.1 
'9    95 

+  548 
+49  4° 

9.14 

/    6.84 
\    6.62 

Ma 

}* 

1.26 
0.64 

JGroombridge  2789.  . 

B.  D.  +  30°  3639... 

19  30.9 

}ai    2-4 

+3°  '8 
+38  15 

9.85 

/     5-57 
(    6.28 

O 

K5 
K5 

5  27 
5  "5 

61  f  Cveni 

B.  D.  +38°  4362... 
Lalande  43492  

21    5.2 
22    12.3 
32  24.5 

23  16.8 

23  44.0 

23  44  9 

+38  19 

+  12   24 

+  57  '2 

+43  33 
+   i  52 
+  2  19 

7.68 
6  97 
9  43 

7.56 
9  09 
8.28 

K 

F8 
(K5) 
F8 
K5 
G 

0.06 
0.83 
o  95 

0.68 
i  40 
o  44 

Hels-Gotha  13170.  . 
(Kriiger  60). 

Lalande  46650  

Lam.  12805.  . 

OF  OBSERVATION. 


77 


TABLE  A  (Continued).    PARALLAXES  DERIVED  WITH  THE  AID  OP  CATALOGUED  PROPER-MOTIONS. 


No. 

Designation. 

Rght 
ascen- 
sion 
1900  o 

Decli- 
nation 
1900.0 

Magni- 
tude. 

Spectrum. 

Proper- 
motion. 

Paral- 
lax. 

Prob- 
able 
error 

P.E. 
from 
obser- 
vations. 

P.E. 
from 
formula. 

Series. 

Compari- 
son-stars. 

Plates. 

Exposures. 

st 

*2 
< 

45 

j;    Geminorum  . 

6h  8™8 

+22°32' 

3  .  2  to  4  .  2 

Ma 

o'.oj 

+OT034 

±oro25 

0^025 

oTo25 

XXXII 

3 

5 

19 

50 

46 
47 

Oj  Geminorum.  }         „ 
a  s  Geminorum.  /  ' 

+32    6 

/     2.85 
\     1.99 

A 
A 

}o.2o{ 

+0.104 

+0.  102 

±0.036 
±0.028 

0.036 

O.O22 

0.036 
0.033 

}  XXXIII 

9 

6 

23 

3-3 

Mean          .... 

+0.103   ±0  029 

O.O27 

0.031 

18 

'     1C     Cl     8 

4-  I  e    en 

a  86 

F8 

I      33 

-0  081 

±0  024 

o  018 

o  028 

XXXIV 

8 

4 

16 

5.0 

4Q 

f  Herculis    ..      !     16  a"?  6-4-al   x"? 

3     OO 

G 

o  60 

+0  101 

±0  024 

0.024 

o  023 

XXXV 

8 

6 

20 

5.1 

5° 

TJ  Herculis  

'6  39-5+39    7 

3.6i 

K 

O.  IO 

+0.014 

±0.066 

0.066 

0.025 

XXXVI 

8 

6 

24 

5-2 

51 

12 

It  Herculis  A  ... 
n  Herculis  EC  . 

j  17  43.6+27  47 

/    3.48 
\     9.68 

G5 

}o.82{ 

+0.024 
+0.051 

±0.028 
±0.040 

0.032 
0.049 

0.024 
0.026 

}  XXXVII 

8 

(I 

27 
23 

5-0 
5-0 

Mean 

i 

+0.038 

±0.036 

0.036 

0.024 

!! 

1 

NOTES  ON  TABLE  A — Continued. 

(44)  Proper-motion  from  A.  G.  Albany.    Measured  because  it  was  on  the  plates  of  Series 

XXXI. 

(45)  Bu.  3239;  /3  1008;  290°,  i  To  (1900);  Physical  pair  in  slow  motion;  Principal  star  vari- 

able; Mag.  3.2  to  4.2 ;  Period  231  days;  also  a  spectroscopic  binary;  Mag.  of  com- 
panion 8.8.  Observed  with  color-screen. 

(46,47)  Castor.  Bu.  4122;  2mo;  224°,  5*6  (1904.)  Combined  mag.  1.58;  Binary;  Period 
about  400  years;  Relative  motion  (1904)  +oTo5O  in  x,  — 0^032  in  y  (from  Lewis's 
diagram).  Both  components  spectroscopic  binaries,  of  period  2.93  and  9.22  days. 
The  companion  of  mag.  9.03  at  164°,  73"  shares  the  proper-motion  and  belongs  to 
the  system.  Observed  with  color-screen. 

(48)  Observed  with  color-screen. 

(49)  Bu.  7717;   22084;    193°.  i'i  (1904);   Mags.  3.0  and  6.5;  Binary;   Period  34.5  years; 

o=  i?4;  companion  not  shown  on  the  plates;  Relative  motion  in  1904  +0^076  in  x, 
— o?os6  in  y;  Masses  approximately  equal  (Lewis).  Observed  with  color-screen. 
Mass  of  companion  given  by  Boss  as  0.43  that  of  primary. 

(50)  Bu.  7738;   Supposed  to  be  a  close  double,  but  undoubtedly  single;   Observed  with 

color-screen. 

(51,  52)  Bu.  8162;  22220;  245°,  32^2  (1904);  Companion  a  close  pair  (A.  Clark  7);  68°, 
i?5  (1904) ;  not  separated  on  the  plates;  Difference  of  magnitude  i.o  (Lewis) ;  which 
makes  the  individual  magnitudes  10.0  and  n.o;  Close  pair  binary;  Period  44  years; 
a  =  1^4;  Wide  pair  a  physical  system  in  slow  motion;  —0^043  in  x,  +0^007  in  y 
(Lewis).  Bright  star  observed  with  color-screen;  faint  companion  outside  it. 


§2.  Reality  of  the  Results.     Negative  Parallaxes. 

There  are  in  all  fifty-five  determinations  of  parallax,  for  forty  stars  or 
pairs  of  stars.  If  these  are  classified  according  to  the  ratio  of  the  observed 
parallax  TT  to  its  probable  error  r,  the  results  are  as  shown  in  Table  35,  p.  78. 
In  the  second  column  the  individual  determinations  are  counted,  and  in 
the  last  the  final  values  for  the  different  stars  or  systems.  The  negative 
parallaxes  are  in  each  case  just  one-fifth  of  the  whole  number.  If  an  equal 
number  of  the  smaller  positive  results  are  assumed  to  be  illusory,  it  follows 
that  60  per  cent,  of  the  stars  observed  have  really  sensible  parallaxes.  The 
largest  negative  parallax  resulting  from  the  observations  is  —  o".o8i.  All  the 


DETERMINATIONS  OF  STELLAR  PARALLAX. 


stars  whose  observed  parallaxes  are  positive  and  numerically  greater  than 
this  (and  also  many  of  the  remainder)  are  therefore  presumably  nearer  us 
than  the  comparison-stars. 


TABLE  35. 


Determi- 
nations. 

Stars. 

T  greater  than  jr.  .  .  . 

"7 
5 
4 
II 

7 

14 

3 
4 

5 
7 

T  between  4r  and  3r 

3r  and  ar. 

ar  and  r  . 

T  and  o 

Total  number  of  po 
r  negative  and  bet  we 

Total  number  of  ne 

sitive  parallaxes  

44 

33 

8 
i 

5 

2 
1 

r  and  ar  

"\r  and  4r.  . 

gative  parallaxes  

ii 

8 

It  is,  however,  noteworthy  that  the  two  largest  negative  parallaxes 
shown  in  table  A  (Nos.  27  and  48),  the  only  two  that  considerably  exceed 
their  probable  errors,  are  derived  from  series  of  but  four  and  five  plates 
(disposed  in  the  latter  case  in  such  a  manner  that  the  weight  of  the  parallax 
is  unusually  small).  These  series  are  insufficient  for  a  reliable  determination 
of  parallax ;  but,  as  circumstances  beyond  the  writer's  control  prevented  their 
extension,  it  seemed  desirable  to  give  their  results  among  the  others,  in 
order  that  the  whole  outcome,  good  and  bad,  of  the  work  might  be  clearly 
exhibited.  It  is,  however,  proper  to  call  attention  here  to  their  relatively 
low  accuracy. 

It  should  be  noticed,  too,  that  the  only  other  series  which  gives  an 
equally  low  weight  for  the  parallax  contributes  another  negative  value  (No. 
34)  and  that  the  three  stars  which  lie  away  from  the  center  of  the  field,  in  a 
position  unfavorable  to  accuracy,  also  give  negative  results  (Nos.  3,  39, 
and  44). 

Of  the  remaining  forty-seven  determinations,  including  all  those  made 
under  conditions  even  reasonably  favorable  to  accuracy  (star  central  and 
weight  of  determination  of  parallax  greater  than  1.5)  only  four  are  negative. 
Nine  out  of  the  eleven  negative  parallaxes  are  less  than  their  mean  errors; 
hence  a  better  formal  representation  of  the  observations  could  be  secured 
by  suppressing  the  parallax  terms  in  the  equations  of  condition  altogether. 
This  seems,  however,  to  be  of  doubtful  propriety,  for  there  is  no  reason  to 
suppose  that  the  relative  parallax  in  these  cases  is  really  exactly  zero,  and 
moreover  this  plan  is  equivalent  to  systematically  rejecting  those  results 
where  the  errors  of  observation  diminish  the  parallax  beyond  a  certain  limit, 
and  would  vitiate  the  mean  values  derived  from  groups  of  observations. 


RESULTS  OF  OBSERVATION.  79 

II.  COMPARISON  WITH  OTHER  OBSERVERS. 

§3.  Description  of  Table  B. 

As  has  already  been  mentioned,  a  large  number  of  the  stars  in  the  above 
list  have  been  observed  elsewhere  for  parallax — the  fact  in  many  cases  not 
being  made  public  until  after  the  present  work  was  under  way.  In  conse- 
quence a  large  amount  of  material  is  available  for  comparison  of  the  results 
of  different  methods  of  observation.  The  determinations  of  the  present  work, 
being,  as  far  as  can  be  discovered  from  internal  evidence,  homogeneous  and 
free  from  sensible  systematic  error,  may  not  unreasonably  be  employed  as  a 
standard,  not  of  absolute  accuracy,  but  of  comparison,  to  which  the  results 
of  others  may  for  the  moment  be  referred. 

Table  B  gives  in  summary  form  the  principal  results  of  modern  observers 
for  the  stars  of  the  present  list.  Such  a  collection  can  make  no  claim  to 
finality  in  the  present  rapidly  growing  state  of  observation;  but  compari- 
son with  a  manuscript  list  kindly  furnished  by  Professor  Kapteyn  in  return 
for  a  summary  of  the  results  of  the  present  work,  and  with  the  extensive 
Catalogue  of  M.  Bigourdan  in  the  Bulletin  Astronomique  (July-December, 
1909),  gives  occasion  for  the  hope  that  few  published  determinations  of  value 
have  been  omitted. 

The  exact  limits  of  exclusion,  especially  for  the  older  observations,  are 
largely  a  matter  of  opinion.  It  is,  however,  improbable  that  moderate  differ- 
ences in  this  respect  would  sensibly  alter  the  conclusions  hereafter  expressed. 

The  first  column  gives  the  current  number  of  the  star  in  table  A ;  the 
second  gives  the  parallax,  and  the  third  the  probable  error,  determined  by 
the  observer  whose  name  follows  in  the  fourth  column.  The  probable  error  is, 
except  when  noted,  that  derived  directly  from  the  residuals  of  the  equations  of 
condition,  the  agreement  of  the  parallaxes  derived  from  various  plates,  or  the 
like — that  is  from  the  "internal "  consistency  of  the  observations  of  the  same 
observer  and  series.  For  the  three  spectroscopic  determinations  the  annexed 
probable  errors  are  rough  estimates  by  the  writer,  based  on  the  data  given 
in  the  notes.  In  the  case  of  Flint,  who  has  applied  systematic  corrections  to 
his  observed  parallaxes,  both  the  observed  and  corrected  values  are  given. 
As  he  makes  no  estimate  of  the  uncertainty  of  these  empirically  derived  cor- 
rections, the  probable  error  of  his  corrected  results  can  not  be  given.  It 
must,  however,  be  greater  than  that  assigned  to  the  observed  parallaxes. 

The  next  column,  headed  Method,  shows  the  general  nature  of  the 
method  of  observation;  H  denoting  observations  made  with  the  heliometer; 
M  results  obtained  with  the  equatorial  and  filar  micrometer;  P,,  those 
derived  photographically  from  series  of  separate  plates;  and  Px  those  from 
plates  on  which  exposures  are  made  at  three  epochs  according  to  Kapteyn 's 
method ;  5  parallaxes  derived  from  spectroscopically  observed  radial  veloci- 
ties in  binary  systems;  Tt,  those  determined  by  means  of  meridian  transits, 
and  T,  by  transits  with  the  equatorial  telescope. 


8o 


DETERMINATIONS  OF   STELLAR  PARALLAX. 


The  column  headed  "Reference"  gives  references  to  the  notes  which 
follow  the  table.  These  show  the  source  from  which  the  tabular  information 
has  been  derived  and  add  occasional  remarks. 

The  last  two  columns  give,  in  thousandths  of  a  second  of  arc,  the  excess 
of  the  corresponding  determination  above  that  of  the  present  work  for  the 
same  star  or  system,  and  its  probable  error,  calculated  by  taking  the  square 
root  of  the  sum  of  the  squares  of  the  probable  errors  determined  by  the  two 
observers  separately  from  the  accordance  of  their  own  observations — the 
values  used  for  the  present  work  being  those  finally  adopted  in  table  A. 

TABI.E  B. — RESULTS  OF  OTHER  OBSERVERS. 


No. 

Parallax. 

Probable 
error. 

Observer. 

Method. 

Reference. 

Difference. 

Probable  error 
of  difference. 

+oTi? 

•toTosg 

Flint 

T, 

i 

+  88 

*  35 

-4-Q,    |Q 

Flint  (corr.) 

+  18 

• 

+  0.292 

*O.O25 

Auwers 

T, 

2 

+  43 

*  30 

+  0-44 

*o.034 

Flint 

T, 

I 

+  190 

*  38 

-4-o  *  I 

Flint  (corr.) 

+  60 

4 

i  "•  y  • 

+0.18 

±0    010 

Peter 

H 

3 

-     7 

±    22 

—  0.03 

±0.044 

Flint 

T, 

l 

-307 

*  49 

—  Lr>    5  « 

Flint  (corr.) 

+  15? 

7 

1   u-54 

+0.037 

±0.020 

Chase 

H 

4 

*       7  J 

+  30 

*  34 

8,9 

—0.04 

±0.040 

Flint 

T, 

I 

-  36 

±   50 

—  O     11 

Flint  (corr.) 

-136 

u.  14 

+0.035 

*o.oi7 

Chase 

ii 

4 

+  39 

*  34 

+0.057 

*o.oi7 

von  Zeipel 

P. 

5 

+  61 

*  3i 

+0.10 

±0.02 

Rambaut 

P, 

6 

+  104 

*  36 

10 

+0.31 

±0.055 

Flint 

T, 

, 

+331 

*  57 

~~  o  02 

Flint  (corr.) 

—      0 

+0.039 

*o.oi8 

Chase 

ii 

4 

+  50 

±  33 

ii 

+0.117 

*o.oig 

Elkin,  Chase 

H 

4 

+  39 

±  27 

11 

+0.101 

±0.026 

Kapteyn 

T, 

7 

+  52 

*  35 

13 

+0.438 

±0.0)0 

Kapteyn 

T, 

7 

+  84 

*  33 

+0.36 

*o.047 

Flint 

T, 

t 

+  16 

*  49 

•4-O    37 

Flint  (corr.) 

+   36 

T          t 
+0.36I 

*0.023 

Jost 

T, 

'8 

+  '7 

*  27 

14 

+o.  167 

*O.O27 

Kapteyn 

T, 

7 

+    4 

*  33 

+0-34 

±0.  110 

Flint 

T, 

i 

+  '77 

±111 

'  * 
•4-O    ^7 

Flint  (corr.) 

+207 

15,  16 

\u.  3/ 

-0.039 

*O.037 

Chase 

H 

4 

-  9° 

*  4' 

17 

+0.000 

*0.025 

Brunnow 

M 

10 

—    10 

±  38 

+0.139 

+O.02 

±0   026 

±0.055 

Kapteyn 
Flint 

?: 

7 
i 

18 

*  39 
*  63 

—  O.OI 

Flint  (corr.) 

—  no 

( 

+0.085 

±0.024 

Jost 

T, 

's 

-  "5 

*  38 

18,  19 

+0.051 

±0.025? 

Belopolsky 

S 

ii 

-  "3 

±  36? 

SI 

+0.40 

^0.065 

Flint 

T, 

l 

+  179 

*  68 

*  fl    11 

Flint  (corr.) 

+209 

1    **  •  ^J 

+0.174 

±0.043 

Elkin 

H 

4 

-  47 

*  47 

36,  37 

+0.038 

±0.053 

Elkin,  Smith 

H 

4 

+  96 

*  67 

3° 

+0.18 

*0.055 

Flint 

T, 

1 

+  85 

*  58 

+  O    17 

Flint  (corr.) 

+  75 



r  v  .  •  / 
+0.17 

*o.oi7 

Chase 

ii 

4 

+  75 

*  33 

1 

RESULTS  OF   OBSERVATION. 


8l 


TABLE  B. — RESULTS  OF  OTHER  OBSERVERS — Continued. 


No. 

Parallax. 

Probable 
error. 

Observer. 

Method. 

Reference. 

Difference. 

Probable  error 
of  difference. 

3'.  32 

+0.353 

±0.014 

Lamp 

II 

12 

+  55 

*  43 

+0.36 

±0.043 

Flint 

T, 

i 

+  62 

*   59 

4-O   :*2 

Flint  (corr.) 

+    22 

i  \j  .  2* 

+0.29 

±0.021 

Kostinsky 

P 

13 

I          ** 

-    8 

±  46 

-|-o.  301 

Bohlin 

P? 

IQ 

4-      :> 

>    41 

+0.282 

±  o  .  004 

Schlesinger 

P, 

26 

T^     y 
-  16 

*r    n* 
*    42 

34-35 

—  O.O2I 

±0.008 

A.  Hall 

M 

14 

-  53 

*     5' 

—  O.O27 

±0.024 

Chase 

H 

4 

-  49 

*   55 

+  0.04 

±0.021 

Kostinsky 

P, 

13 

+    8 

*   54 

+0.064 

±0.040 

Neander 

P, 

"5 

+  32 

±  64 

37.38 

+0.326 

±0.035 

Kapteyn 

P, 

16 

-  58 

±  41 

+O.2I 

±0.029 

Flint 

T, 

I 

-'74 

±  36 

+O.2I 

Flint  (corr.) 

—  174 

+0.270 

±0.010 

A.  Hall 

M 

'4 

1  /*? 
-114 

±  23 

+0.340 

±0.029 

Peter 

H 

17 

-  44 

±  36 

+0.293 

±0.007 

Bergstrand 

P, 

18 

—  91 

±    22 

+0.38 

±0.015 

Kostinsky 

P8 

13 

±    26 

+0.320 

±0.028 

Jost 

T, 

8 

-  64 

*  35 

+O.29I 

±0.005 

Chase 

H 

20 

—  93 

±    22 

+0.23 

±0.035 

Abetti 

T, 

21 

-'54 

*    4' 

39 

+o.  116 

±0.021 

Kapteyn 

Pa 

16 

+  151 

±    36 

+0.04 

±0.021 

Kostinsky 

P, 

13 

+  75 

*    36 

40 

+0.140 

±0.106 

Chase 

H 

4,28 

+  119 

±108 

4' 

+0.249 

±0.010 

Barnard 

M 

22 

—     9 

*  at 

+0.248 

±0.009 

Schlesinger 

P, 

27 

—   10 

±    21 

43 

+0.031 

±0.016 

Chase 

TT 

4 

-     6 

±  26 

43 

+oT39 

±0^092 

Flint 

Tt 

I 

+  '79 

•*=  95 

+0.23 

Flint  (corr.) 

+    '9 

+0.200 

±0.084 

Elkin 

H 

4 

I       •  y 
—     II 

*  87 

46,  47 

—  o.  14 

±0.041 

Flint 

T, 

I 

-243 

*  50 

—  o.  17 

Flint  (corr.) 

—  272 

+O.O22 

±0.010 

Smith 

H 

23 

i  J 

-  81 

*  3' 

+0.07 

±0.03? 

Curtis 

s 

24 

-  33 

±  42? 

48 

+0.11 

±0.028 

Flint 

T, 

I 

+  191 

*  37 

+O.O5 

Flint  (corr.) 

+  131 

+0.090 

±0.024 

Chase 

ii 

4 

1      •  J  • 

+  17" 

*  34 

49 

+0.172 

±0.029 

Smith 

H 

4 

+  71 

*  37 

+  0.  II 

±0.025? 

Lewis 

s 

25 

+     9 

*  33? 

50 

+O.2I 

±0.035 

Flint 

T, 

I 

+  196 

*  75 

+o.  15 

Flint  (corr.) 

+  136 

5'.5* 

+O.  122 

±0.028 

Chase 

H 

4 

1     *  J»* 

+  84 

±  46 

NOTES  TO  TABLE  B. 

i)  Publications  of  the  Washburn  Observatory,  vol.  xi  (1902). 

The  parallaxes  resulting  directly  from  the  observations  are  found  by  the  author  to 
require  systematic  corrections,  depending  on  the  apparent  difference  of  magnitude 
between  the  parallax-star  and  the  mean  of  the  comparison-stars.  These  corrections 
were  determined  empirically  by  comparison  of  the  parallaxes  derived  with  the  aid 
of  individual  comparison-stars  of  different  magnitude.  Their  amount  varies  with 
the  R.  A.,  the  spurious  parallax  of  a  star  one  magnitude  fainter  than  the  comparison- 
stars  being  +o7i7  at  oh  and  3h,  +oTio  at  2ih  and  6h,  +oTos  at  i93h  and  7^h, 
oToo  at  i8h  and  ioh,  and  — oTos  at  I7h  and  I3h. 

The  probable  error  of  these  systematic  corrections  must  be  considerable.  Both  the 
uncorrected  and  corrected  values  are  given  in  the  table. 


82  DETERMINATIONS  OF   STELLAR  PARALLAX. 

(2)  Berlin  A  had.  Abhandlungen,  1867.     (Math.),  page  18. 

(3)  Astr.  Nach.,  3533  (1898). 

(4)  Transactions  of  the  Astronomical  Observatory  of  Yale  University,  vol.  II,  Part  I  (1906). 

The  annexed  probable  errors  are  those  resulting  directly  from  the  observations 
of  each  star,  as  given  in  the  body  of  the  work,  and  not  those  given  in  the  final 
table  of  results  (pages  196-198)  which  are  increased  to  allow  for  the  effect  of 
a  presumed  systematic  error  of  =*=  o!o3o  for  an  average  determination  (except 
for  No.  30,  where  the  mean  result  of  several  series,  given  on  page  198,  is  taken). 

(5)  Astr.  Nach.,  4188  (1907).    Hour-angles  for  morning  and  evening  observations  differ 

by  7  hours. 

(6)  Monthly  Notices.,  LXX,  page  325.    (1910.)    Photographs  taken  near  the  meridian. 

Monthly  Notices,  LXVII,  page  259.     (1907.) 

(7)  Annalen  der  Sternwarte  in  Leiden,  Bd.  vn,  p.  119.    (1897.) 

(8)  Vierteljahrsschrijt.     1906,  p.  146. 

(10)  Dunsink  Observations  II,  p.  19.    (1873.) 
(n)  Astr.  Nach.,  3510.    (1898.) 

(12)  Astr.  Nach.,  2676  and  2807.     (1885,  1887.) 

(13)  Publ.  de  I'Obs.    Central  Nicolas  (Poulkowa)  sdrie  II,  vol.  xvn  (1905),  pp.  129  ff., 

140-141.  Hour-angles  at  morning  and  evening  observations  differ  by  6  hours 
or  more. 

(14)  Washington  Obs.  1883.    Appendix  n.    Observations  with  26-inch  equatorial. 

(15)  Astronomische  J ahresbericht  1907,  p.  289.    Components  not  separated. 

(16)  Publ.  of  the  Astronomical  Laboratory  at  Groningcn.    No.  10,  pp.  48,  58.     From 

photographs  by  Donner  at  Helsingfors.  Hour-angles  for  morning  and  evening 
observation  differ  by  7§  hours. 

(17)  From  manuscript  list  sent  by  Prof .  Kapteyn.     Original  reference  undiscovcrable. 

(18)  Astr.  Nach.,  3999.    (1904.)    Atmospheric  dispersion  determined  and  allowed  for. 

(19)  Astr.  Nach.,  4365.    (1909.)    From  two  epochs  only;  dispersion  taken  into  account. 

Bohlin  gives,  from  A"  cos  o,  r  =-f-oT296;  from  A5,  it  = 0^306;  from  both 
together  T  =+0^251.  The  mean  of  the  first  two  has  been  taken. 

(20)  Astronomical  Journal,  593.     (1907). 

(21)  Astr.  Nach.,  4270.     (1908.) 

(22)  Monthly  Notices,  ucvm,  p.  637.     (1908.)     Observations  with  Yerkes  4o-inch 

equatorial. 

(23)  Astronomical  Journal,  594.      (1907.) 

(24)  Astrophysical  Journal,  vol.  23,  p.  351.    (1906.)    From  radial  velocities  of  centers 

of  mass  of  the  two  components  and  Doberck's  orbit,  T  =0^05.  The  two 
alternative  orbits  given  by  Doberck  (Astr.  Nach.,  3970)  lead  with  the  same 
radial  velocities  to  parallaxes  of  oTo6  and  oT  1 1 .  The  mean  of  the  three  has 
been  taken. 

(25)  Memoirs  of  the  Royal  Astronomical  Society,  vol.  LVI,  p.  471.    From  changes  in  the 

radial  velocity  of  the  bright  component,  r  =0^14.  If  Boss's  result,  that 
the  mass  of  the  fainter  component  is  0.43  times  that  of  the  brighter,  is  taken 
instead  of  Lewis's  conclusion  that  their  masses  are  equal,  the  parallax  is  reduced 
to  0^09.  The  mean  of  the  two  has  been  taken. 

(26)  Aslr.  Nach.,  4365.     (1909.) 

(27)  Monthly  Notices,  LXVIH,  p.  637.    (1908.)    From  photographs  with  Yerkes  4o-inch 

equatorial.  Hour-angles  limited  and  isochromatic  plates  used  to  minimize 
atmospheric  dispersion. 

(28)  Corrected  to —oTo47=*=  0^046.    Yale  Transactions,  vol.  11,  p.  295.    Corrected  differ- 

ence from  result  of  presentwork  —  oTo68  =*=o?o52.  Received  too  late  for  incor- 
poration into  the  body  of  this  work. 


RESULTS  OF   OBSERVATION.  83 

§4.  Search  for  Systematic  Errors. 

Systematic  error  in  determinations  of  parallax  may  be  of  two  kinds: 

(1)  There  may  be  a  constant  tendency  toward  too  great  or  too  small 
values  for  stars  of  a  given  class.    Examples  of  this  are  the  personal  equation, 
depending  both  on  magnitude  and  right  ascension,  detected  and  empirically 
corrected  by  Flint  in  his  earlier  work,  and  the  influence  of  atmospheric  disper- 
sion, varying  with  the  color  of  the  star,  which  is  to  be  feared  in  photographic 
work  when  all  the  exposures  are  not  made  at  the  same  hour-angle. 

(2)  In  addition  to  this,  and  even  in  its  absence,  there  may  be  causes 
of  error  at  work  which  bring  about  discrepancies  between  the  results  of 
different  series  of  observations  on  the  same  star,  greater  on  the  average 
than  would  be  expected  on  the  basis  of  the  probable  errors  derived  from  the 
internal  agreement  of  the  observations  of  each  series,  but  varying  from  series 
to  series  in  an  apparently  random  manner,  or  at  least  without  any  clearly 
discernible  law.    Examples  may  be  found  in  some  of  the  work  of  the  Yale 
heliometer. 

It  is  worthy  of  remark  that  discordances  of  this  type  may  be  expected 
(to  a  greater  or  less  extent)  whenever  the  observations  are  confined  to  the 
minimum  number  of  parallactic  epochs  necessary  to  separate  the  unknowns ; 
for  any  errors,  whether  instrumental  or  personal,  which  vary  slowly  with 
the  time,  will  be  practically  constant  during  the  few  weeks  within  which 
the  observations  at  any  epoch  lie,  and  so  will  affect  the  values  of  the  un- 
knowns without  perceptibly  affecting  the  agreement  of  the  observations  at 
any  one  epoch,  or  increasing  the  residuals  from  which  the  probable  error 
is  derived.  Errors  of  this  type  are  much  less  serious  than  those  first  de- 
scribed, since  their  action  is  rather  to  diminish  the  accuracy  of  the  resulting 
parallax  than  to  vitiate  it. 

If  no  systematic  errors  are  present  (or,  at  least,  if  they  are  identical  in 
sign  and  magnitude  in  the  two  groups  of  observations  compared)  the  results 
of  the  two  must  agree,  on  the  average,  within  the  limits  of  error  defined  by 
the  probable  errors  derived  from  the  internal  agreement  of  the  observations 
of  each  series.  This  may  be  tested  in  two  ways: 

(a)  The  probable  error  of  one  difference,  deduced  in  the  ordinary  way 
from  the  mean-square  or  numerical  average  values  of  these  differences, 
must  agree  with  that  already  derived. 

(b)  The  actual  distribution  of  the  ratios  of  these  differences  to  their 
probable  errors  must  conform  to  that  demanded  by  theory. 

When,  as  in  the  present  case,  the  probable  errors  of  the  individual  deter- 
minations vary  through  a  wide  range,  the  second  method  is  preferable  to 
the  first  (which  amounts  to  giving  the  greatest  weight  to  the  poorest  obser- 
vations, especially  if  the  mean  of  squares  is  taken) .  It  has  also  the  advantage 
that  it  shows  whether  any  discordance  is  due  to  a  general  prevalence  of 
differences  exceeding  their  probable  errors,  or  to  a  few  large  discrepancies. 


84 


DETERMINATIONS  OF   STELLAR   PARALLAX. 


Applying  these  methods  to  the  data  of  table  B,  the  results  are  as  shown  in 
table  36 :  The  photographic  observations  have  been  divided  into  two  groups : 
(a)  those  in  which  precautions  are  known  to  have  been  taken  to  eliminate  or 
determine  the  influence  of  atmospheric  dispersion,*  and  (6)  those  in  which 
they  do  not  appear  to  have  been  considered  (mostly  of  an  earlier  date,  before 
the  importance  of  the  matter  was  fully  realized).  The  latter,  along  with 
Flint's  results,  have  been  put  by  themselves,  on  account  of  the  possible  or 
probable  presence  of  systematic  error. 

TABLE  36- 


Method. 

Excess  above  present  work. 

Ratio  of  excess  to  its  probable  error. 

Average  with- 
out regard  to 
sign. 

Average  with 
regard  to  sign. 

otoi 

1  tO  2 

2  to  3 

3104 

Over  4 

Total. 

Heliometer  

o"i>5<)     ±  0^042 
0.048     ±0.035 
0.045      ±o-°33 
0.018     ±0.037 
o  052     ±0.035 

0.052     ±0.038 

+o?oi2    *oToio 
—  0.026  '.  ±0.016 
—  0.002  ;  ±0.015 

—  O.OI2      ±O.02I 
+O.OOI      ±O.OI2 

+0.001     ±0.006 

f 

3 

3 
4 

18 

20 

4 

2 

6 

8 

6J 

2 

3 

i 

1} 

18 
5 
5 
3 
9 

40 
40 

8 

'5l 
|M 

>$} 

Micrometer  

Photography  (a)  

' 

Spectroscopic  
Transits  

3 

"i 
•  3 

2 

5 
7 

5 

i 

5 
5 

i 

2 
>) 

All  together  

1 

5} 

2 

Theory.  . 

Photography  (i)  

0.049     ±0.042 

o.  144     ±0.059 
0.109   (±0.002) 

-(-0.032    ±0.015 

+0.048    ±0.015 
+0.028  (±0.023^ 

Flint: 
Uncorrected  

Corrected  

Theory 

J 

i 

! 

The  probable  errors  given  in  the  third  column  of  this  table  are  the  simple 
means  of  those  of  the  individual  differences  given  in  table  B,  except  in  the  last 
line  but  one,  which  will  be  explained  later.  In  the  absence  of  systematic  error 
each  of  these  quantities  should  be  less  by  15  per  cent  than  that  in  the  preceding 
column.  Those  in  the  fifth  column  are  obtained  by  dividing  the  former  by 
the  square  root  of  the  number  of  observations  combined  to  form  the  mean 
and  are  (very  nearly)  what  the  probable  errors  of  the  means  in  the  fourth 
column  would  be,  in  the  absence  of  systematic  error. 

The  two  stars  (Nos.  26-27  and  48)  whose  parallaxes  are  worst  deter- 
mined in  the  present  work  (see  §2)  are  given  half  weight  in  forming  these 
means  and  in  the  counts  detailed  in  the  following  columns,  since  in  these 
cases  the  use  of  the  present  work  as  a  standard  of  comparison,  though  neces- 
sary for  homogeneity,  is  otherwise  by  no  means  desirable. 

All  the  results  in  which  there  was  not  good  reason  to  suspect  definite 
systematic  errors  of  known  origin  are  collected  in  the  upper  part  of  table  36. 
It  is  clear  at  a  glance  that  the  systematic  differences  between  these  and  the 
present  work  must  be  small.  The  mean  differences  (taking  account  of  sign) 
never  greatly  exceed  their  probable  errors,  and  the  magnitudes  of  the  indi- 

*The  wort  of  Bergstrand,  Bohlin,  Rarabaut,  and  Schlesingrr. 


RESULTS   OF   OBSERVATION. 


vidual  discordances  are  distributed  in  general  agreement  with  the  law  of 
chance,  when  the  very  small  numbers  in  some  of  the  groups  are  considered. 

When  the  various  groups  are  combined  there  appears  distinct  evidence 
of  something  more  than  random  error.  This  does  not,  however,  consist  of  a 
general  increase  in  the  number  of  cases  in  which  the  discordances  of  indi- 
vidual determinations  exceed  their  probable  error  (which  is  what  might  be 
expected  if  sources  of  systematic  difference  were  generally  or  frequently  at 
work).  Instead,  there  is  a  small  group  of  large  discordances,  exceeding  four 
times  their  computed  probable  error,  while  the  smaller  discordances  are  dis- 
tributed in  close  agreement  with  theory. 

These  cases  deserve  special  consideration.  One  of  them  arises  from 
the  large  negative  parallax  found  for  star  No.  48  (7  Serpentis) — which  has 
already  been  pointed  out  as  the  weakest  of  all  the  determinations  of  the 
present  work,  and  which  for  comparison  purposes  has  been  given  half  weight. 
The  other  three  large  discordances  are  all  for  the  same  star,  61  Cygni  (Nos. 
37,  38),  and  arise  from  the  disagreement  of  the  result  of  the  present  work 
with  those  of  the  exceptionally  long  and  careful  series  of  Hall,  Bergstrand, 
and  Chase,  which  agree  closely  inter  se.  Here  there  can  be  little  doubt  that 
the  parallax  found  in  the  present  work  is  too  great  by  o''o8  or  thereabouts. 

As  these  stars  are  the  brightest  which  were  observed  without  the  color- 
screen*,  and  are  photographically  2.5  and  2  magnitudes  brighter  than  the 
average  of  their  comparison-stars,  it  is  very  probable  that  this  is  a  case  of 
guiding  error — a  supposition  which  is  confirmed  by  the  fact  that  the  parallax 
of  the  brighter  component  comes  out  o''o4  greater  than  that  of  the  fainter. 

It  is,  however,  hardly  fair  to  count  this  one  error  as  three— especially 
since,  by  comparison  with  other  observers,  this  star  has  already  contributed 
three  discordances,  all  exceeding  their  probable  errors,  to  our  list.  If  they 
are  reduced  to  one,  the  observed  and  theoretical  distribution  of  the  dis- 
cordances, in  terms  of  their  probable  errors,  compare  as  shown  in  table  37. 

The  agreement  with  theory  is  now  appar- 
ently very  close.  However,  the  mean-square 
value  of  all  the  discordances,  in  terms  of  their 
probable  errors,  is  on  this  calculation  1.77. 
Reducing  this  itself  to  a  probable  error,  by 
multiplying  by  0.6745,  we  find  that  the  prob- 
able error,  deduced  from  the  agreement  of  the 
results  of  different  observers  with  the  present 
work,  is  on  the  average  1.19  times  that  derived 
from  the  internal  agreement  of  the  observa- 
tions of  the  series  in  question — which  would  indicate  the  existence  of  some 
source  of  systematic  discordance,  whose  average  amount  corresponds  to  a 
probable  error  0.65  times  that  deduced  from  the  internal  agreement  of 

'Except  No.  3,  which  was  observed  only  because  it  happened  to  be  shown  on  the  plates  of  another  star. 


TABLE  37. 


Ratio. 

Observed. 

Theory. 

o  to  i 

18 

'9 

I    to  2 

llj 

12 

2    to   3 

5 

5 

3  to  4 

2 

'i 

4  to  5.1 

ii 

? 

Total  

38 

38 

86  DETERMINATIONS  OF  STELLAR  PARALLAX. 

the  observations.  The  average  value  of  the  latter  for  the  cases  in  question 
is  ±oTo38.  The  average  systematic  error,  expressed  as  a  probable  error,  is 
therefore  =fcoTo25,  including  the  combined  errors  of  both  observers. 

Approaching  the  matter  in  a  somewhat  different  way,  but  with  the  same 
restrictions  as  above,  the  average  value  of  one  discordance,  without  regard 
to  sign,  is  found  to  be  o!o49,  to  which  corresponds  a  probable  error  0.845 
times  as  great,  or  =*=oTc>4i4.  The  average,  without  regard  to  sign,  of  the 
probable  errors  computed  from  the  internal  agreement  of  the  observations 
is  =fco?o385.  This  would  indicate  a  systematic  discordance  of  amount  cor- 
responding to  a  probable  error  of  =±=  0^015  for  the  difference  between  the 
results  of  two  observers. 

It  might  be  argued  that  mean-square  values  instead  of  the  arithmetical 
means  ought  to  be  taken ;  but  this  would  be  equivalent  to  giving  the  greatest 
weight  to  the  poorest  determinations,  unless  some  means  are  adopted  to 
reduce  all  to  a  uniform  standard  of  weight;  and  this  has  already  been  done 
in  the  procedure  first  followed.  If  the  three  large  discordances  for  61  Cygni 
are  counted  separately,  the  value  of  the  systematic  error  comes  out  ±0^032 
by  the  first  method  and  =fcoTo22  by  the  second;  but,  for  reasons  already 
stated,  these  are  probably  somewhat  too  great. 

The  mean  of  all  these  determinations  is  ±0^024,  which  may  be  adopted 
as  the  probable  error  corresponding  to  the  average  influence  of  the  combined 
systematic  errors  of  other  observers  and  the  present  work  on  the  difference 
of  their  results. 

If  these  errors  were  uniformly  divided  among  all  the  observations,  the 
average  systematic  error  of  one  observed  parallax  would  be  ±0^017.  Those 
of  some  observers  are  doubtless  greater  than  this,  and  of  others  less.  For 
example,  the  observers  at  Yale,  from  comparison  of  the  results  of  successive 
series  of  observations  on  the  same  star,  estimate  that  the  systematic  error  of 
a  parallax  derived  from  a  single  such  series  is  =*=  0^030.* 

The  average  systematic  difference  from  the  present  work  (determined 
by  the  second  of  the  methods  described  above,  without  weighting  down  the 
comparisons  for  61  Cygni)  is  ±0^028  for  the  heliometer  observations  (all  but 
two  of  which  were  made  at  Yale)  and  ±oToi6  for  all  the  others. 

The  latter  value  corresponds  to  an  average  systematic  error  of  ±o!on 
in  the  individual  determinations  of  parallax.  If  we  assume  this  as  the  sys- 
tematic error  of  the  observations  of  the  present  work  which  were  compared 
with  the  heliometer  results,  and  likewise  of  the  two  of  these  not  obtained  at 
Yale,  the  average  systematic  error  of  the  Yale  results  comes  out  ±0^027. 

If  we  take  into  account  the  fact  that  four  of  the  latterf  are  the  means 
of  two  or  four  series,  the  average  systematic  error  of  a  single  series  becomes 
±0^029,  accidentally  in  almost  perfect  agreement  with  the  estimate  of  the 
observers. 


•Yale  Transactions,  vol.  H,  part  I,  p.  194.  fKor  stars  Nos.  7,  30,  37,  and  46. 


RESULTS  OP  OBSERVATION.  87 

The  last  four  lines  of  table  36  illustrate  the  effects  of  systemtic  error. 
Of  Flint's  uncorrected  results,  only  one-eighth  differ  from  those  of  the 
present  work  by  less  than  the  computed  probable  errors.  The  average  prob- 
able error  derived  from  the  differences  themselves  is  more  than  double  that 
deduced  from  the  internal  agreement  of  the  observations.  Large  systematic 
errors  are  clearly  present.  After  the  corrections  deduced  by  the  observer 
have  been  applied,  things  are  much  better.  It  would  be  unfair  in  this  case 
to  demand  agreement  within  the  limits  set  when  the  unknown  probable 
errors  of  the  systematic  corrections  are  neglected.  It  is  better  to  derive  the 
probable  error  of  one  difference  from  their  average  value,  and  this  has  been 
done  in  the  last  line  but  one  of  the  table.  It  appears  that  the  probable  error 
of  one  of  the  corrected  results  is,  on  the  average,  rather  more  than  half  as 
great  again,  and  the  weight  rather  less  than  half  as  great,  as  the  internal 
agreement  of  the  observations  would  indicate.  If  the  probable  errors  of  the 
individual  differences  given  in  table  B  are  increased  in  the  same  proportion, 
the  distribution  of  the  ratios  of  one  to  the  other  is  that  given  toward  the  end 
of  the  line  in  question,  in  table  36. 

There  is  now  little  evidence  of  outstanding  systematic  error.  By  the 
application  of  the  corrections,  the  mean  difference  (regarding  signs)  between 
the  two  sets  of  results  considered  is  reduced  to  little  more  than  half  its  initial 
value,  and  becomes  comparable  with  its  probable  error,  and  the  distribution 
of  the  individual  discordances  is  in  fair  agreement  with  that  predicted  by 
theory  and  given  in  the  last  line  of  the  table. 

There  is  also  evidence,  of  a  somewhat  different  kind,  that  systematic 
error  exists  in  those  photographic  results  in  which  the  influence  of  atmos- 
pheric dispersion  was  not  eliminated,  as  in  the  present  work  and  that  of 
other  recent  observers.  Here  neither  the  distribution  of  the  individual 
discordances  nor  their  average  numerical  value  shows  signs  that  anything  is 
amiss;  but  the  photographic  parallaxes  come  out  too  great,  on  the  average, 
by  0^032,  which  is  more  than  twice  its  probable  error. 

This  in  itself  would  be  hardly  too  much  to  attribute  to  chance;  but  it 
agrees  in  sign  and  magnitude  with  the  error  which  might  be  anticipated. 
The  "  average  spectrum,"  if  the  phrase  may  be  used,  of  the  stars  under  inves- 
tigation is  K,  and  that  of  the  comparison-stars  about  FS,  so  that  the  effective 
refraction  constant  is  probably  sensibly  less  for  the  former.  If  the  difference 
is  6/3,  the  formulae  of  Chapter  III,  §7  (taking  as  average  conditions  ^  =  45°, 
6  =  45°,  t=  —  3h  for  the  morning  and  + 3''  for  the  evening  observations),  show 
that  an  error— 0.65/3  will  appear  in  the  relative  parallax  deduced  from 
photographs.  The  observed  discordance  therefore  demands  that  5/3  shall 
be  about  —  o''o5,  which  is  of  the  order  of  magnitude  indicated  both  by  our 
knowledge  of  the  dispersion  of  air  and  by  direct  photographic  investigations. 

The  results  of  this  discussion  may  be  summarized  as  follows : 

Systematic  errors  of  the  first  kind — i.  e.,  constant  errors  depending  on 
magnitude,  spectrum,  and  the  like — exist  in  Flint's  work  (where  they  have 


88 


DETERMINATIONS  OF   STELLAR   PARALLAX. 


been  effectively  corrected  by  the  author)  and  in  the  photographic  observations 
made  at  varying  hour-angles,  but  are  absent,  or  at  least  very  small,  in  the 
work  of  other  observers.  Those  of  the  second  kind  —  which  amount,  so  far 
as  can  be  determined,  to  an  increase  of  the  probable  error  of  observation  — 
are  generally  present,  but  very  small.  In  the  observations  made  at  Yale 
they  appear  to  be  somewhat  less  than  was  estimated  by  the  observers  them- 
selves. For  all  other  observations,  including  the  present  work,  their  average 
influence  on  the  parallax  corresponds  to  a  probable  error  only  very  slightly 
exceeding  0*0  1. 

The  influence  of  this  systematic  error  may  be  included  in  the  probable 
errors  given  in  table  A  by  increasing  those  less  than  0*020  by  0*003,  those 
between  0*020  to  0*033  by  0*02,  and  those  greater  than  this  by  oTooi. 

III.  COMPARISON  WITH  KAPTEYNTS  FORMULAE. 
§5.  Parallax  of  the  Comparison-Stars. 

According  to  Kapteyn,*  the  mean  parallax  of  all  the  stars  of  visual 
magnitude  m  is 

7TH,  =  0*0160  (0.75)  "-»  (0 

while  that  of  the  group  of  magnitude  m  and  proper-motion  /z  is 

*W  =  (0.87)   —  "   \/T^  (2) 


where  A  =0.0753,  P=  I-2°  f°r  stars  °f  tne  fifst  spectral  type,  and  A  =0.0316, 
£=1.47  for  those  of  the  second.  (In  deriving  these  formulae,  all  the  faint 
stars  of  large  proper-motion  were  considered  to  be  of  the  second  type  —  an 
assumption  invariably  verified  upon  investigation.) 

Applying  the  first  of  these  to  the  comparison-stars  (grouped  according 
to  their  photometric  magnitude,  as  in  Chapter  IV,  §3)  the  results  are  as 
follows  : 

TABLE  38. 


No.  of 
stars. 

Mean 
magnitude. 

Computed 
parallax. 

Difference 
from  mean. 

Mean  observed 
parallax. 

25 

9 

106 

5* 

7.32 
8.64 

95" 
10.35 

+0^0008 
+0.0065 
+0.0051 
+0.0039 

-f-O  OOS7 

+oToo4 

+0.001 
—  O.OOI 
—  O.OO2 

+0.006 

—O.OOI 
—  O.OOJ 

+0.003 

The  mean  parallax  of  the  comparison-stars  may  therefore  be  taken  as 
+0*006.  The  observed  mean  parallaxes  of  the  groups  of  different  magni- 
tude, relative  to  the  whole,  agree  with  the  theoretical  values  very  closely. 

For  the  six  comparison-stars  showing  the  clearest  evidence  of  proper- 
motion  (listed  in  Chapter  IV,  §7),  the  average  magnitude  and  proper- 

•Publkations  of  the  Astronomical  Laboratory  of  Croningen,  No.  8,  p.  24;  revised,  No.  n,  p.  18. 


RESULTS   OF   OBSERVATION. 


89 


motion  are  9.66  and  0*32;  the  corresponding  parallax  is  0^023.  The  mean 
parallax  of  the  comparison-stars  for  the  fields  in  which  one  of  these  stars  lie 
should  therefore  be  taken  as  oTooS. 

From  the  data  of  page  62  it  appears  that  the  probable  error  of  the  mean 
parallax  of  the  six  or  eight  comparison-stars  of  a  given  field  is  about  one- 
third  of  its  value.  It  is  therefore  possible  to  pass  from  the  observed  relative 
parallaxes  to  the  absolute  parallaxes  by  adding  the  amounts  just  given,  with- 
out sensibly  increasing  the  probable  error  derived  from  the  observations. 

§6.  Data  for  the  Individual  Parallax-Stars. 

The  individual  results  for  the  stars  specially  observed  for  parallax  are 
as  follows:  The  observed  parallaxes  have  been  increased  by  the  computed 
parallax  of  the  comparison-stars,  to  render  them  comparable  with  the  theo- 
retical values.  The  latter  have  been  derived  from  the  formula  (2),  consider- 
ing all  spectra  from  O  to  FS,  inclusive,  as  of  Type  i,  and  those  from  F8  to  M 
as  of  Type  n.  For  star  No.  36  the  proper-motion  derived  from  the  plates 
has  been  employed  in  the  absence  of  better  data.  Otherwise  the  data  are 
those  of  table  A. 

For  double  stars  and  pairs  with  common  proper-motion,  the  mean  of 
the  observed  parallaxes  and  the  computed  parallax  for  the  brighter  compo- 
nent are  given.  For  the  long-period  variable  Mira  (No.  5)  the  computed 
parallaxes  corresponding  to  the  average  maximum  and  minimum  brightness 
are  given.  The  former,  which  is  nearer  the  observed  value,  has  been  used 
in  the  discussion  of  the  results. 

TABLE  39.    COMPARISON  WITH  KAPTEYN'S  FORMULA. 


No. 

Mae  !pr°Pcr 
lag'  ;motion 

Sp. 

Observed 
parallax. 

Computed 
parallax. 

No. 

Mag. 

Proper 
motion 

Sp. 

Observed 
parallax. 

Computed 
parallax. 

i 

2.4 

0.56    Fj        +oTo88 

+oTno 

36,  27 

6.8  !    0.68 

G 

_0,0 

+oTo6o 

2 

7-7 

2.80 

Ma       +0.258 

+0.141 

28,  29 

6.8      0.48 

G?        +0.046 

+0.049 

3 

6.0 

0.03     A 

—0.018 

+0.006 

30 

7.8       1.36  i  Ma       +0.101 

+0.085 

4 

3.6 

1.24    F8 

+0.193 

+o.  140 

3'.  32 

93 

2.27     KS        +0.304 

+0.098 

5 

(3.o\ 
I  9-JJ 

0.24 

Md 

+o.  142 

/  +0.051 

\    +0.021 

33 
34.  35 

r* 

1.26 
0.64 

Ma       +0.082       +0.067 
K         +0.038       +0.060 

6 

3.8 

0.17 

Mb 

+0.091 

+0.038 

36 

9.8 

0.03      O             —0.012 

+0.005 

7 

2.  1 

o.oi     B8 

+0.013 

+0.005 

37.38 

5-6 

5.27 

KS 

+0.390 

+0.290 

8.9 

8.3 

1.38    G 

+0.002 

+0.o8o 

39 

7-7 

0.06 

K 

—0.029 

+O.OIO 

10 

8.3 

2.  2O 

G2 

—0.005 

+0.109 

40 

7.0 

0.83     F8        +0.029      +0.068 

n 

6.8 

0.68 

G 

+0.084 

+0.o6l 

4" 

94 

o  95 

(KS)'   +0.264  :   +0.053 

12 

65 

0.90 

F8 

+0.057 

+0.078 

42 

7-6 

0.68 

F8        +0.045  '    +0.055 

'3 

74 

4.78     K 

+0.350 

+0.211 

43 

9  i 

1.40 

Ma       +0.219       +0.073 

"4 

8.6 

4.40 

KS       +0.169 

+0.169 

44 

8.3 

0.44 

G         —0.014  •     +o  °35 

15.  16 

6.4 

o  74 

K     '     +0.058 

+O.O7O 

45 

35 

0.07 

Ma       +0.040  '     +0.021 

•7 

6.5 

7.04    G         +0.106 

+0.315 

46,47 

2.O 

O.2O 

A 

+o.  109  i    +0.051 

18,  19 

36 

0.56 

F         +0.070 

+0.095 

48 

3-9 

'33 

F8 

-0.075  |    +0.144 

20 

10.0       1.89 

(65)     +O.IH 

+0.079 

49 

3.0 

0.60 

G 

+0.107 

+0.096 

21 

8.3       2.32 

KS       +0.227 

+0.114 

50 

36 

o.  10 

K 

+O.O2O 

+0.022 

22,23 

94       1-38 

K 

+0.030 

+0.070 

5',  52 

35 

0.82 

GS 

+0.044         +O.IO9 

24.  25 

8.9 

3  75 

G5 

+0.036 

+0.147 

< 

DETERMINATIONS   OF   STELLAR   PARALLAX. 


§7.  Comparison  by  Groups.  Systematic  Differences  for  Different  Spectral  Types. 

Grouping  together  those  stars  whose  computed  parallaxes  lie  between 
specified  limits,*  the  observed  and  computed  values  compare  as  follows, 
the  general  agreement  being  very  good: 

TABLE  40. 


Limits  of 
computed  parallax. 

Mean  i 
Observed. 

>arallax. 
Computed. 

No.  of 

stars. 

>0'20 

0*20  to  oTio 
o.  10  to  0.07 
0.07  to  0.05 
<oTo5 

0^282 
o.  104 
o.  106 
0.082 
0.015 

0^272 
o.  132 
0.082 
0.059 

O.O2I 

3 
9 

10 

9 
9 

Altogether  

O  OQ} 

o  089 

Taking  means  for  groups  of  stars  of  similar  magnitude,  proper-motion, 
or  spectral  type,  the  results  are  shown  in  tables  41  and  42. 

TABLE  41. 


Limits  of— 

No.  of 

stars. 

Mean 
mag. 

Mean 
proper- 

Mean  parallax. 

Ratio. 

Computed 
ratio. 

motion. 

Observed. 

Formula. 

Magnitude. 

2.0  to    4.0 

12 

32 

oT49 

+0^070 

+oTo73 

1.0 

1.1 

5.6  to    7.0 

9 

6.4 

1.83 

+0.079 

+o.  no 

0.7 

0.8 

7.  i  to    8.0 
8.1  to    9.0 

6 
6 

Ki 

'  75 
2.41 

+o.  126 
+0.079 

+0.095 
+0.109 

'  3 
0.7 

"  3 
0.9 

9.0  to  10.0 

7 

94 

1.31 

+0.143 

+0.063 

23 

'3 

Proper-motion. 

Over     3  To 

5 

74 

5ro5 

+0?2IO 

+0.226 

0.9 

i.i 

3.0  to  1.5 

5 

8-7 

2.30 

+0.179 

+0.108 

'•7 

"-4 

1.5  to  i  .0 

7 

73 

'34 

+0.079 

+0.094 

0.8 

1  .0 

i.o  to  0.7 

5 

66 

0.85 

+0.090 

+0.076 

1.2 

0.9 

0.7  to  0.4 

9 

5-8 

o  59 

+0.046 

+0.069 

0-7 

0.6 

Under  0.4 

9 

4.6 

o.  10 

+0.040 

+0.023 

'7 

1-4 

TABLE  42. 


Spectrum. 

No.  of 
stars. 

Mean 
magnitude. 

Mean 
proper-motion. 

Mean  parallax. 

Ratio. 

Observed. 

Formula. 

O  to  F5  \ 
Type  i     / 

6 

43 

oT23 

+oTo42 

+0:045 

09 

F8 

5 

57 

1.00 

+0.050 

+0.097 

05 

G,  G2 

7 

6.9 

1.86 

+0.032 

+0.109 

0-3 

£ 

6 

u 

1:3 

+0.059 
+0.078 

+0.096 
+0.074 

0.6 
1.1 

K5                    5 

8.3 

3.04 

+0.271 

+0.145 

"  9 

M 

7 

6-3 

1.04 

+0-"33 

+0.068 

'9 

Type  ii 

34 

67 

I'.ta 

0*102 

0:096 

1.1 

*It  will  not  do  to  group  them  according  to  the  observed  parallaxes,  for  the  groups  of  largest  parallax 
will  then  contain  a  disproportionate  number  of  stars  whose  parallaxes  are  greater  than  the  average  for  stars 
of  the  same  magnitude  and  proper-motion,  and  also  of  those  whose  parallaxes  are  increased  by  errors  of 
observation,  and  similar  systematic  errors  of  opposite  sign  will  appear  in  the  groups  of  smallest  parallax. 


RESULTS   OF   OBSERVATION. 


The  last  column  but  one  in  table  41,  and  the  last  column  in  table  42 
give  the  ratio  of  the  mean  observed  parallax  for  each  group  to  the  mean  of 
the  values  predicted  by  Kapteyn's  formula. 

For  the  different  spectral  types  these  ratios  vary  in  a  strikingly  sys- 
tematic fashion.  The  first  type  stars  (O  to  FS)  are  too  few  in  number  to 
permit  of  separation  into  sub-groups.  Among  the  remaining  stars,  for  which 
Kapteyn's  "second-type"  formula  was  used  the  ratio  of  the  observed  to  the 
computed  parallax  increases  rapidly  and  almost  regularly  with  increasing 
redness  from  less  than  one-half  for  Type  G  to  more  than  double  for  Type  M. 
If  the  stars  under  discussion  are  separated  according  to  magnitude  (above 
and  below  7.0)  or  proper-motion  (greater  or  less  than  i''o)  and  similar  ratios 
are  taken  (combining  adjacent  spectral  classes  to  get  enough  stars)  the  results 
are  as  follows : 

TABLE  43. 


Spectrum. 

Average 
proper- 
motion. 

Ratio. 

No.  of 
stars. 

Average 
magnitude. 

Ratio. 

No.  of 
stars. 

F8-G2 

2?64 

o  3 

5 

5-3 

°  5 

7 

G5-K     :      2.94 

i.o      |        4 

54 

0.7 

5 

K5-M 

2.6} 

'•7 

8 

4.0 

'.? 

4 

F-G2 

0.68 

0.6 

7 

7-9 

0.2 

5 

GS-K 

0.47 

0.6 

6 

8.7 

1  .O 

5 

K5-M 

0.35 

33 

4 

8-7 

2.0 

8 

The  phenomenon,  therefore,  appears  to  persist  throughout  a  consider- 
able range  of  magnitude  and  proper-motion. 

For  the  average  of  all  these  stars  together  the  observed  and  computed 
values  are  in  close  agreement,  thus  confirming  Kapteyn's  formula  for  them 
as  a  whole.  It  is,  however,  evident  that,  with  the  more  detailed  spectro- 
scopic  data  now  available,  the  accuracy  of  the  prediction  of  parallax  can 
be  considerably  increased  by  taking  these  differences  into  account.  For 
example,  the  ratios  of  the  observed  and  computed  parallaxes  of  the  groups 
of  stars  of  similar  magnitude  or  proper-motion,  given  in  table  41,  vary 
through  a  wide  range,  in  a  very  irregular  fashion.  If  for  each  of  these  groups 
the  mean  of  the  ratios  corresponding  to  the  spectral  types  of  the  individual 
stars  is  taken,  the  values  given  in  the  last  column  of  that  table,  under  the 
head  "Computed  ratio"  are  obtained.  It  is  clear  that  a  large  part  of  the 
irregularity  of  the  observed  ratios  arises  from  the  irregular  distribution  of 
stars  of  the  different  spectral  types. 

The  material  here  discussed  is  not  a  large  enough  part  of  the  whole  avail- 
able information  concerning  stellar  parallax  to  warrant  the  derivation  of 
corrections  to  Kapteyn's  formula  from  its  results  alone.  The  remarkable 
increase  in  parallax  with  increasing  redness  is  clearly  demonstrated  among 
the  stars  of  considerable  proper-motion;  but  it  is  quite  uncertain  whether 
it  will  be  found  to  hold  good  among  those  of  small  proper-motion. 


92 


DETERMINATIONS   OF   STELLAR   PARALLAX. 


It  is  obvious  that  the  use  of  a  single  formula  for  all  the  yellow  and  red 
stars  may  give  rise  to  serious  systematic  errors;  but  the  adoption  of  the 
factors  of  correction  here  derived,  for  groups  of  stars  differing  greatly  in 
proper-motion  from  those  used  in  deriving  them,  might  lead  to  equally 
erroneous  results. 

Further  investigation  of  the  matter  is  eminently  desirable,  especially  the 
determination  of  the  spectral  types  of  all  the  faint  stars  which  have  been 
observed  for  parallax  and  a  study  of  the  parallactic  motions  of  the  brighter 
stars  of  each  spectral  type  separately. 

IV.  ASTROPHYSICAL  DATA. 

§8.  Brightness  and  Cross  Velocities  of  the  Individual  Stars. 

Table  45  shows  what  information  can  be  derived  from  the  results  of  the 
present  work  concerning  the  actual  brightness  of  the  stars  observed  for  paral- 
lax and  their  velocities  at  right  angles  to  the  line  of  sight.  The  former  of 
these  —  following  Kapteyn  —  is  expressed  in  terms  of  the  "absolute  magni- 
tude," i.  e.,  the  magnitude  which  the  star  would  seem  to  have  if  placed  at 
such  a  distance  that  its  parallax  was  one-tenth  of  a  second  of  arc.  This  is 
found  from  the  observed  magnitude  m  by  the  equation 


The  absolute  magnitude  of  the  sun,  on  this  scale,  is  31.58  magnitudes 
fainter  than  its  apparent  stellar  magnitude.  According  to  the  recent  determi- 
nation of  Prof.  W.  H.  Pickering,*  the  latter  is  —26.83,  which  would  make 
the  Sun's  absolute  magnitude  4.75.  Earlier  determinations  make  it  con- 
siderably lower:  for  example,  Kapteyn  adopts  the  value  5.5. 

The  actual  brightness,  in  terms  of  the  Sun,  of  a  star  of  given  absolute 
magnitude,  according  to  those  two  determinations,  is  shown  in  table  44, 
the  light  decreasing  tenfold  for  every  2.5  magnitudes. 

TABLE  44- 


Absolute. 

Light,  in  terms  of  Sun. 

Absolute. 

Light,  in  terms  of  Sun. 

magnitude. 

Kapteyn. 

Pickering. 

magnitude. 

Kapteyn. 

Pickering. 

3.50 

15.85 

7  95 

4  35 

3  .16 

'  59 

2-75 

12.59 

6.31 

4.50 

2.51 

1.26 

Voo 

10   OO 

5.01 

475              2.00 

1  .00 

3  25 

7  95 

3  98 

5.00 

1.59              0.80 

35° 

6.31 

3.16 

5  25 

1.26              0.63 

3-75 

5  .01 

2.51 

5.50 

i.oo              0.50 

4  oo 

3°« 

2.00                575 

II 

0.80 

0.40 

The  cross-velocities  (that  is,  the  component  of  the  velocity,  relative  to 
the  vSun,  which  is  perpendicular  to  the  line  of  sight)  are  given  in  kilometers 

per  second  and  are  computed  by  the  formula  Tr=4-74  -,  where  /u  is  the  annual 

7T 

proper-motion  on  a  great  circle. 


•Harvard  Annals,  LXI,  part  I,  p.  69. 


RESULTS   OF   OBSERVATION. 

TABLE  45. 


93 


Star. 

Sp. 

No. 

Absolute  magnitude  correspond- 
ing to  the  parallax. 

Cross  velocity  corresponding    to 
the  parallax. 

Decreased 
by  proba-    Observed, 
ble  error. 

Increased 
by  proba- 
ble error. 

Decreased 
by  proba- 
ble error. 
Km.  /sec. 

Observed. 
Km.  /sec. 

Increased 
by  proba- 
ble error. 
Km.  /sec. 

+30°3639... 

ft  Persei  
26  Androm.  . 
Lai.  27743... 
a!  Gemin.  .  . 
a,  Gemin  .  .  . 

y  Virginis.  .< 
0  Cass  

O 

B8 
A 
A? 
A 
A 

F 
F 
F5 

F8 

F8 

F8 
F8 
F8 

G 
G 

G2 

G 
G 

G 
G 
G 

(G5) 
G5 
K 
K 
G5 

36t 

B1 
& 

47*t 

18* 
19* 
i* 

/    4* 
\Comp. 

12 

40 

* 

8 
9 

10 

n 
'7 

26 

8- 

20 
25 
24 
28 

29     A 
..    .    B 

(4-8) 

O.  I 
3    a 

| 

-2.3 

3.5 

1  .2 

6 

i 

M 

"  9 
19 

t  5 

4.8 
8.6 
4.0 

(1.2) 

45 

30 
2.2 

2.9 
29 
2.1 

6.1 
8.8 
1.1 

4.3 
5.8 

u  } 
H  } 

2.5 
5.3    I 

9.0       / 
5-9 
5-5 
6.7 

n 

60 
40 

34 

134 

(550) 
132 

g 

38 
31 

30 

77 

135 
74 

7 

28 
25 

28 

55 
77 
49 

rj  Cass 

Gr.  1646  
Lal.434Q2... 
Lai.  45755... 
•ySerpentis.  . 

Lai.  6888... 
Lal.688Q.... 
Lai.  7443.... 
Gr.  884 

(0.8) 
(1.4) 

£8      ) 
6.4      / 

(3-1) 
6.9 
7-' 

(3300) 

205 
(1200) 

A 

59 

5.8 

6.4 
6.6 

50 
430 

38 
315 

Gr.  1830  

Lai.  27742  .  .  . 
Lara.  32805.  . 
f  Here  

& 

10.6 

7-7 
7-9 
6.7 

6.3 
6.6 
3-o 
95 
10.5 

10.2 

»    i 

8.2        j 

8.4     j 

$ 

6.3    J 

no 

22 

66 
3" 

36 

48 

62 

38 
105 
35 

2.6 

9.6 
4.8 
5-0 
52 
4.8 

(-1.0) 
(4-5) 
(5-5) 

IO.O 

3-6 

4-7 

32 

10  2 
67 
6.9 
6.1 
5.7 
6.0 
1.7 
82 
9.2 

10.1 
53 
64 
6.8 
7.0 
4.7 
4.5 

34 

103 
1180 

72 
(480) 

68 
1  20 

27 

81 
495 

48 
88 

65 
58 

216 
80 

Berl.  A  4099. 
A.  Oe.  14320. 
A.  Oe.  14318. 
W.B.i5b7i6. 
VV.B.i5h72o. 
(O22Q8).. 

M  Here    . 

G5 

51*  A 

»{? 

'3 
\l 

22 
23 

34 

ft 

50* 
'4 

21 

3" 
32 

J 

/  41 
\Comp. 

« 

10 

;  33 

[iit 

Companion.. 

Lai.  2  1185... 
83  Leonis.  .< 

Berl.  8.5072. 
Berl.  B.  5073 

K 
K 

K 

K 

Gr.2789...{ 
+38°4362  .  .  . 

K 
K 
K 
K 

K5 
K5 

KS 

K5 
K5 
K5 

(Kj) 

Ma 
Md 
Mb 
Ma 
Ma 
Ma 
Ma 

0.1 

M 

10.1 
11.7 
12  4 
85 
9.2 
11.8 
13.1 

98 
2.5to10.4 
3.1  to  3.  9 
78 
87 
10  8 
1  3  to2  3 
| 

3-3 

10.  1 

10.3 

12.0         1 
12.7         1 

8.6      \ 
93      < 

12.  0 

'33      J 

99 
2.9  to  10.8 
i  4.0  to   4.8 
8.1 
99 

11.0 

2.3  to  3.3 

24 

124 
48 

35 
63 
17 

51 
8 
9 
64 
73 
30 
8 

6 

112 

44 
31 
60 
it 

48 
6 
6 

55 
4" 
27 
5 

Lai.  21258... 
Lai.  25372... 
Pos.  M.2i64/ 
(2  2398)...  1 

61  Cygni..  | 

H-G  13170) 
(Kruger  6o),f 

Gr.  34 

9-5 
99 
11.4 
12.  i 

8.4 

,?:6 
12.9 

9.7 
1.9  109.8 

1.  9  to  2.7 

7-5 
5-3 
10.5 
—0.8  too.  a 

138 
53 
4i 

67 
.8 

55 
10 
16 
"6 
350 
34 

21 

o  Ceti 

p  Persei  .... 
W.  B.  |7>>322 
Lam.  32805  . 
Lai.  46650.  .  . 
i)  Gemin.  .  .  . 

'Naked-eye  stars  (above  the  fourth  magnitude). 


tProper-motion  less  than  0*4. 


94  DETERMINATIONS   OF   STELLAR   PARALLAX. 

In  tabulating  the  results  the  stars  are  numbered  as  in  table  A,  but 
arranged  in  order  of  spectral  type.  The  mean  observed  parallax  of  the  two 
stars  of  a  physical  pair  is  adopted  for  both.  For  certain  double  stars  the 
magnitudes  of  companions,  not  shown  on  the  plates,  are  included,  and  for 
some  variables  the  maximum  and  minimum  magnitudes  are  given.  To  show 
the  extent  of  the  uncertainty  due  to  errors  of  observation,  the  values  result 
ing  when  the  observed  parallax  is  increased  or  decreased  by  its  probable 
error  are  given  on  each  side  of  those  corresponding  to  the  observed  parallax 
which  are  printed  in  heavier  type.  When  the  parallax  used  for  computation 
is  less  than  oToi,  the  corresponding  entry  in  the  table  is  inclosed  in  paren- 
theses ;  and  when  it  is  negative  the  space  is  left  blank. 

In  examining  these  results,  it  should  be  remembered  that  it  is  as  likely 
as  not  that  the  true  values  for  any  given  star  lie  outside  the  range  of  those 
given  in  the  table,  and  that  in  some  cases  they  are  doubtless  very  consider- 
ably outside  this  range.  It  must  also  be  borne  in  mind  that  the  stars  included 
in  this  list  are  but  few  in  number  and  have  been  selected  from  the  general 
mass  according  to  definite  apparent  characteristics — practically  all  having 
large,  or  at  least  considerable,  proper-motions,  and  all  lying  within  the  limits 
of  magnitude  set  by  the  necessity  of  correct  exposure  of  the  plates  (with  or 
without  the  color-screen). 

For  example,  all  but  one  of  the  bright  stars,  observed  with  the  color- 
screen,  appear  to  be  actually  brighter  than  the  Sun,  while  few  of  the  more 
numerous  stars  observed  without  it  (which  are  almost  all  invisible  to  the  naked 
eye)  appear  to  equal  the  Sun  in  luminosity.  There  is  no  doubt  that  this  is 
due  almost  entirely  to  the  fact  that  the  latter  were  selected  on  account  of 
their  large  proper-motion,  and  hence  represent  stars  much  nearer  us  than  the 
average  of  those  of  the  same  magnitude,  and  necessarily  really  much  fainter. 

In  any  discussion  it  is  therefore  desirable  to  separate  these  two  groups. 
Is  has  also  seemed  best  to  exclude  from  the  latter  group  the  four  stars  whose 
proper-motion  is  less  than  0^40.  Three  of  these  (Nos.  3,  36,  and  39),  with 
proper-motions  less  than  oTio  and  parallaxes  apparently  insensible,  are 
clearly  in  no  way  comparable  with  the  remaining  stars,  while  the  fourth  is 
the  long  period  variable  Mira,  which  ought  obviously  to  be  kept  by  itself. 

§9.  Means  for  Different  Spectral  Types. 

It  is  a  matter  of  some  difficulty  to  find  really  representative  mean  values 
for  any  group  of  stars.  The  results  derived  from  the  observed  parallaxes  of 
each  star  separately  are  affected  by  the  errors  of  observation.  Since  a  given 
decrease  in  the  assumed  parallax  of  a  star  increases  its  computed  velocity 
and  brightness  more  than  an  equal  increase  in  the  assumed  parallax  dimin- 
ishes them,  the  mean  of  a  number  of  such  results  may  be  expected  to  come 
out  too  great;  and  if  any  of  the  observed  parallaxes  are  negative,  no  satis- 
factory mean  can  be  found  in  this  way. 


RESULTS   OF   OBSERVATION. 


95 


If,  on  the  other  hand,  the  means  of  the  observed  magnitudes,  proper- 
motions,  and  parallaxes  are  taken,  and  the  corresponding  cross- velocity  and 
luminosity  found,  the  results  may  be  expected  to  be  too  small,  on  account 
of  the  neglect  of  the  real  departures  of  the  parallax  of  individual  stars  from 
the  mean.*  The  true  values  will  usually  lie  between  the  mean  values  found 
in  these  two  ways. 

Grouping  the  stars  in  this  way,  according  to  spectral  type,  the  results 
are  as  shown  in  table  46.  The  parallax  of  the  comparison-stars  has  been 
allowed  for  and,  as  usual,  only  the  brighter  star  of  a  physical  pair  is  counted. 
A  blank  in  the  last  column  denotes  that  some  of  the  observed  parallaxes  are 
negative,  and  no  mean  of  the  individual  values  can  be  taken. 

TABLE  46. 
Naked-Eye  Stars. 


Spectrum. 

No.  of 
stars. 

Mean 
mag. 

Mean 
proper- 
motion. 

Mean 
observed 
parallax. 

Correspond- 
ing absolute 
magnitude. 

Corre- 
sponding 
velocity. 
Km.  /sec. 

Mean  of 
individual 
absolute 
magni- 
tudes. 

Mean  of 
individual 
velocities. 
Km.  /sec. 

B8  to  F;..  . 
F8  to  M  .  .  . 

4 
7 

2.6 
V7 

oT33 
0.62 

o'.ojo 
0.060 

1.8 

2.6 

22 
4Q 

1.2 

20 



Fainter  Stars  of  Large  Proper-motion. 


F8. 

a 

o  80 

c    2 

86 

G,  G2  

6 

7    ? 

2    O7 

O  O2O 

4O 

508 

Gs 

2 

8  6 

2    OJ. 

7  6 

1  5  1 

7    3. 

208 

K 

I    88 

78 

67 

K?  .  . 

1 

8  * 

a    O4 

o  271 

m    C 

IO    3. 

M 

8  4 

i  66 

48 

The  differences  in  actual  brightness  and  velocity  between  the  two  groups 
of  naked-eye  stars  are  closely  parallel  to  those  in  their  apparent  brightness 
and  proper-motion,  and  may  be  due  largely  to  arbitrary  selection  of  the 
stars,  some  of  which  were  put  on  the  working  list  for  quite  different  reasons 
from  others. 

The  faint  stars  of  large  proper-motion,  on  the  other  hand,  are  apparently 
fairly  homogeneous,  the  mean  magnitude  and  proper-motion  for  all  the 
spectral  sub-groups  except  the  first  being  nearly  the  same.  The  discordant 
values  for  the  stars  of  spectrum  F8,  which  average  a  magnitude  brighter  than 
the  others  and  have  little  more  than  one-third  of  their  mean  proper-motion, 
are  undoubtedly  due  to  the  manner  of  their  selection.  In  preparing  the 
working  list,  stars  below  the  sixth  magnitude,  and  with  proper-motion  less 
than  i"  were  included  only  (i)  if  their  spectrum  was  given  as  A  in  the  Draper 
Catalogue;  (2)  in  the  case  of  pairs  with  common  proper-motion.  The  latter 
are  distributed  almost  uniformly  among  the  different  spectral  types ;  but  the 
former  (although  the  later  observations  at  Harvard  modify  the  original  esti- 

*In  small  groups,  accidental  irregularities  may  mask  this  general  tendency. 


96  DETERMINATIONS   OF   STELLAR  PARALLAX. 

mates  of  the  spectrum)  furnish  all  the  cases  of  spectrum  F8  and  one  G.* 
The  small  average  proper-motion  of  the  first  group  is  therefore  due  to  the 
adoption  of  a  different  standard  of  admission;  while  their  greater  average 
brightness  means  only  that  none  are  included  which  are  too  faint  to  appear 
in  the  Draper  Catalogue. 

The  other  five  groups,  though  very  similar  in  apparent  magnitude  and 
proper  motion,  show  systematic  differences  in  real  brightness  and  velocity 
of  so  marked  a  character  that  they  must  have  some  real  significance. 

§  10.  Possible  Explanations  of  the  Differences. 

It  is  worthy  of  especial  attention  that  if  we  assume  that  either  the 
absolute  magnitudes  or  the  cross  velocities  given  in  table  46  are  character- 
istic of  all  (or  at  least  of  the  large  majority)  of  the  stars  of  the  corresponding 
spectral  type,  then  the  observed  distribution  of  the  other  tabular  quantity 
(cross-velocity  or  absolute  magnitude)  and  of  the  average  parallaxes  becomes 
a  necessary  consequence  of  the  manner  of  selection  of  the  stars. 

Suppose,  for  example,  that,  as  indicated  by  the  absolute  magnitudes  of 
the  table,  stars  of  Type  G  are  on  the  average  2.5  magnitudes  (or  ten  times) 
brighter  than  those  of  Type  K,  and  these  again  as  much  brighter  than  those 
of  Type  M.  Then,  since  our  stars  have  been  so  chosen  that  the  average  appar- 
ent brightness  of  all  three  groups  is  nearly  the  same,  those  stars  of  Type  G, 
which  appear  in  our  list,  must  on  the  average  be  some  three  times  as  far  off 
as  those  of  Type  K,  and  those  again  at  three  times  the  distance  of  those  of 
Type  M ;  and  their  mean  parallaxes  will  be  to  one  another  in  the  inverse  ratio. 

Since  they  have  also  been  selected  so  that  the  mean  proper-motion  of 
the  different  groups  is  nearly  the  same,  the  average  cross-velocity  of  those 
stars  which  pass  the  conditions  of  admission  must  be  about  three  times  as 
great  for  each  type  as  for  the  following. 

This  is  roughly  what  is  shown  in  table  46,  the  differences  being  due  to 
departures  of  the  tabular  numbers  from  the  simple  relations  assumed  in  the 
above  illustration. 

In  just  the  same  way  it  follows  that  if  the  stars  of  Type  G  are  actually 
moving  faster,  on  the  average,  than  those  of  Type  K,  the  method  of  selection 
compels  us  to  choose  brighter  stars,  on  the  average,  in  the  first  case  than  in 
the  second. 

The  observed  differences  in  brightness  and  velocity  can  not  both  be 
results  of  the  method  of  selection ;  but  either  one  may  (not  must)  be  so,  if 
the  other  represents  a  real  characteristic  of  the  different  spectral  types. 

Among  the  various  hypotheses  thus  suggested  there  can  be  little  doubt 
which  is  the  most  plausible.  Recent  researches  have  established  a  very 
strong  presumption  that  the  spectral  type  of  a  star  is  intimately  connected 
with  its  surface  temperature  (the  evidence  being  particularly  convincing  for 
just  the  range  of  spectral  types  under  discussion). 

*No.  ii.     Groombridgc  884. 


RESULTS  OF   OBSERVATION.  97 

The  sequence  of  Types  G,  K,  M  is  almost  certainly  one  of  decreasing 
temperature,  and  therefore  of  diminishing  surface-brightness,  and  the  marked 
diminution  in  the  average  luminosity  of  the  stars  from  type  to  type  is  just 
what  might  be  expected. 

On  the  other  hand,  it  is  very  hard  to  see  how  the  velocity  of  a  star  in 
space  can  be  a  function  of  its  temperature,  especially  to  the  enormous  extent 
demanded  by  the  observations,  if  they  are  to  be  explained  in  this  way. 

Certain  other  facts  confirm  the  former  explanation  of  the  observations. 
The  stars  of  Type  F8,  on  account  of  the  peculiar  method  of  selection  described 
above,  are  apparently,  on  the  average,  considerably  brighter  and  much  more 
slowly  moving  than  the  closely  related  stars  of  Type  G.  The  difference  in 
the  observed  mean  parallaxes  is,  however,  such  as  to  make  the  real  average 
brightness  of  the  two  groups  come  out  nearly  equal,  while  exaggerating  that 
between  the  mean  velocities.  Though  the  number  of  stars  involved  is  small, 
this  may  be  taken  as  confirmatory  evidence  that  brightness,  rather  than 
velocity,  is  the  principal  point  of  similarity  between  adjacent  spectral  types 
and  of  difference  between  those  widely  separated. 

Again,  the  spectroscopic  determinations  of  radial  velocity  show  no  such 
marked  progression  with  the  spectral  type  as  is  exhibited  above,  and  what 
there  is  has  the  opposite  sense— the  mean  velocity,  after  allowance  is  made 
for  the  solar  motion,  being  distinctly  less  for  Type  B  than  for  the  others,  and 
probably  increasing  slowly  from  A  to  M.* 

(The  stars  with  which  such  observations  deal  are  for  the  most  part, 
however,  so  very  different  in  intrinsic  brightness  from  those  here  considered 
that  this  argument  is  by  itself  less  conclusive  than  it  might  at  first  appear.) 

All  things  considered,  it  may  be  regarded  as  probable  that  the  differences 
in  absolute  magnitude  (or  actual  luminosity)  between  the  stars  of  different 
spectral  types,  revealed  in  table  46,  are  real  and  typical,  while  those  in 
mean  parallax  and  cross-velocity  are  consequences  of  this,  together  with 
the  way  in  which  the  stars  were  selected  for  observation. 

The  remarkable  differences  from  Kapteyn's  formula,  described  in  §6, 
(page  91)  may  likewise  be  attributed  to  this  cause. 

§11.  Bearing  on  Stellar  Evolution. 

The  work  of  Scheiner  and  Wilsing  at  Potsdam  makes  it  possible  to  esti- 
mate what  part  of  the  differences  in  brightness  between  the  stars  of  different 
spectral  types  is  due  to  temperature  alone.  They  find  that  the  distribution 
of  brightness  in  the  visual  spectrum  of  a  large  number  of  stars  agrees  closely 

c  x"s 
with  Planck's  formula  JA  =  — ^f~       -  ;  and  with  the  aid  of  this  formula  they 

AT 

e      —    i 
determine  their  effective  temperatures. 

*Kapteyn,  Astrophysical  Journal,  vol.  xxxi,  p.  260  (April  1910). 


98 


DETERMINATIONS  OF   STELLAR   PARALLAX. 


If  the  actual  light  emission  of  the  stars  is  also  approximately  in  agree- 
ment with  the  formula  (which  seems  reasonable)  the  relative  brightness 
(expressed  in  stellar  magnitudes)  of  stars  of  the  same  diameter,  but  of  differ- 
ent spectral  types  and  temperatures,  should  on  their  data*  be  as  follows : 

TABLB  47. 


Spectral  type. 

Absolute 
temp. 

(C.) 

Relative  brightness 
for  wave-length. 

Potsdam. 

Harvard. 

O.6OOM 

0.500,1 

o  400,4 

la  i 
Ha 
Ila-IIIa 
III 

A 
G 
K 
M 

9600° 
5400 
4000 
3200 

oTo 

2.2 

39. 

56 

0"?0 
3.6 

11 

O^O 

M 

5-7 

8.2 

On  this  hypothesis,  the  lower  temperature  accounts  for  nearly  two  mag- 
nitudes of  the  difference  of  absolute  magnitude  between  Types  G  and  K,  or 
K  and  M,  leaving  little  to  be  explained  by  differences  in  the  actual  size  of  the 
stars.  The  latter,  however,  appears  to  be  somewhat  smaller,  and  the  density 
presumably  greater,  for  those  of  lower  temperature.  It  is  probable  that  these 
stars  are  in  the  later  stages  of  evolution,  having  passed  their  maximum  temp- 
erature, and  that  the  reddest  ones  are  approaching  extinction.  Evidence 
that  some  of  them  are  of  considerable  density  will  be  given  later. 

It  should  be  particularly  noticed  that  the  foregoing  remarks  apply  only 
to  stars  whose  actual  luminosity  is  comparable  with  or  less  than  that  of  the 
Sun.  The  majority  of  the  naked-eye  stars  of  these  spectral  types  (and  prob- 
ably of  those  of  any  given  visual  magnitude)  have  small  proper-motions 
and  are  presumably  remote  and  of  great  intrinsic  luminosity — as  is  shown  by 
direct  measurement  for  such  stars  as  Capella,  Arcturus,  and  Antares.  These 
may  reasonably  be  supposed  to  be  stars  in  a  much  earlier  stage  of  evolution, 
of  small  density  and  rising  temperature,  owing  their  high  luminosity  to  their 
great  superficial  area;  and,  if  this  hypothesis  is  sound,  the  reddest  among 
them  are  in  the  most  primitive  condition. 

Red  stars  (of  Types  KS  and  M)  may  therefore  be  supposed  to  represent 
two  widely  different  stages  of  evolution,  one  early  and  one  very  late.  Most 
if  not  all  of  those  which  are  conspicuous  to  the  naked  eye  belong  to  the  former 
class ;  while  the  latter  are  to  be  found  exclusively  among  the  fainter  stars  of 
large  proper-motion. 

§  12.  Masses  of  Binary  Stars. 

Six  of  the  stars  observed  for  parallax  are  binaries  for  which  more  or  less 
satisfactory  orbits  have  been  computed.  The  following  table  gives  for  these 
the  semi-major  axis  a  and  period  t,  with  the  authority  from  which  they  are 
taken;  the  sum  of  the  masses  of  the  components,  computed  by  the  equation 

a3 
m  =  -—— ,  and  on  each  side  of  it  the  values  obtained  by  increasing  or  decreas- 


*Potsdam  Publications,  vol.  xix,  part  I,  p.  67;  the  assumed  value  of  c,  being  14600. 


RESULTS   OF   OBSERVATION. 


99 


ing  the  observed  parallax*  by  its  probable  error;  and  finally  the  "hypothet- 
ical parallax"  p  obtained  from  the  above  equation  on  the  assumption  that 
the  mass  is  in  all  cases  2.4  times  that  of  the  Sun,  the  excess  of  the  observed 
parallax  above  this,  and  the  probable  error  of  the  latter. 

TABLE  48. 


No. 

Star. 

a 

/ 
years. 

Authority. 

Mass,  corresponding  to 
the  parallax. 

Comparison  of 
hypothetical  and 
observed  paral- 
laxes. 

Decreased 
by  its 
probable 
error. 

Ob- 
served. 

Increased 
by  its 
probable 
error. 

4 
18,  19 

29 

46,  47 
49 
52 

tf  Cassiopeia; 
y  Virginis 
OX  298 
a  Geminorum 
f  Herculis 
MHerculisBC 

8T5i 
3.90 
0.88 

5-75 
1.38 

'•45 

333 

182 

56 
347 
34-5 
43  5 

Lewis  (i) 
Lewis  (2) 
Celoria  (6) 
Doberck  (4) 
Doolittle  (5) 
Lewis  (3) 

2.2 

21  .2 

It 
(3100) 

1.6 

53 
23 
12 
1.8 
19.0 

1  .2 
2.  1 

1  .0 
0.7 

1.0 

3.- 

p      *-p      p.  e. 

oTi68+oTo25±o?oig 
0.091—0.021*0.026 
0.045+0.001  ±0.015 
o  .  087+0  .  022  ±  o  .  029 
0.098+0.009*0.024 
0.088—  0.044*0.036 

(i)  Mem.  R.  A.  S.,  LVI,  p.  16.       (2)  Ibid,  p.  339.       (3)  Ibid,  p.  506.       (4)  Astronom.  Nach.,  3970. 
(5)  Astronomical  Jour.  460  (vol.  20,  p.  25).        (6)  Astronom.  Nach.,  2843. 

The  mean  of  the  masses  corresponding  to  the  observed  parallaxes  is 
5.2  times  that  of  the  Sun.  Excluding  the  last  star,  for  which  the  errors  of 
observation  are  unusually  large,  this  is  reduced  to  2.4  times  the  Sun's  mass. 

The  mean-square  difference  between  the  hypothetical  parallaxes,  com- 
puted with  this  latter  value  of  the  mass,  and  the  observed  values  is  0^024, 
while  the  mean-square  probable  error  of  one  such  difference  is  ±0^025,  so 
that  a  mean-square  difference  of  oTo37  might  have  been  expected.  This 
remarkably  close  agreement  is  doubtless  due  to  chance;  but  it  is  clear  that 
the  present  work,  though  showing  conclusively  that  the  average  mass  of 
these  systems  considerably  exceeds  that  of  the  Sun,  does  not  enable  us  to 
arrange  them  in  order  of  mass. 

According  to  Professor  Boss,  the  ratio  of  the  mass  of  the  companion  to 
the  primary,  in  three  of  these  systems  (Nos.  4,  18,  and  49)  averages  0.83. 
It  appears,  therefore,  that  on  the  average  the  principal  star  in  one  of  these 
systems  is  rather  more  massive  than  the  Sun,  and  the  companion  nearly 
equal  to  the  latter. 

There  are  also  three  slow  binaries  on  the  list,  whose  relative  motion, 
though  distinctly  curved,  does  not  yet  cover  a  long  enough  arc  to  permit 
the  calculation  of  even  an  approximate  orbit.  It  does  not  appear  to  be 
generally  known  that  in  such  cases  an  inferior  limit  to  the  mass  of  each 
system,  and  a  close  approximation  to  the  actual  mean  mass  of  a  number  of 
them,  can  be  obtained  if  the  parallaxes  are  known. 

The  observations,  in  such  a  case,  enable  us  to  find  the  coordinates, 
velocity,  and  acceleration  of  one  star  relative  to  the  other,  as  projected  on  a 
plane  tangent  to  the  celestial  sphere,  for  any  convenient  instant  near  the 
middle  of  the  interval  which  they  cover. 

"Corrected  for  the  probable  parallax  of  the  comparison-stars. 


100 


DETERMINATIONS  OP   STELLAR  PARALLAX. 


Since  the  acceleration  must  be  directed  toward  the  principal  star,  the 
motion  of  the  companion  may  be  represented  by  the  expressions 


where  p0  is  the  position  angle  at  the  time,  t0;  s  and  p  the  distance  and  posi- 
tion angle  at  the  time  /;  and  the  terms  involving  the  cube  of  the  time  are 
neglected. 

Now,  according  to  the  law  of  gravitation, 


where  r  is  the  distance  between  the  two  stars.  If  the  astronomical  unit,  the 
year,  and  the  Sun's  mass  are  chosen  as  units,  the  constant  K  is  (2*)',  or 
39-478. 

If  ia  is  the  angle  which  the  line  joining  the  stars  makes  with  the  line 

d'x 
of  sight  at  the  instant  t0>  then  at  this  moment  r  =  a  cosec  i0  and  -7-7  =  —  2c, 

a'c 


whence  (>«,+«;,)  sinsi0=- 


19.74 
and  V  is  the  parallax,  this  becomes 


If  a  and  c  are  expressed  in  seconds  of  arc, 


sin-V0 


a'c 


19.747^ 

The  mass  of  the  system  is  thus  determined,  except  as  regards  the  factor 
sin3«0-  The  second  member  of  the  above  equation  is  evidently  the  minimum 
value  of  the  mass.  In  any  individual  system  more  can  not  be  said  until  the 
elements  of  the  orbit  can  be  determined  ;  but  in  the  mean  of  a  large  number 
of  cases  (since  there  is  no  reason  to  suppose  any  connection  between  the 
direction  of  the  line  joining  the  stars  and  the  line  of  sight)  the  average  value 
of  sin3i0  can  be  found  on  principles  of  geometrical  probability  —  the  reasoning 
being  identical  with  that  familiar  in  the  kindred  case  of  spectroscopic  binaries. 
The  probability  that  sin  i  is  less  than  any  given  limit  sin  ia  is  i  —  cos  i0,  and 

the  theoretical  mean  value  of  sin'/0  is  -    or  0.589.    The  actual  mean  is  likely 

to  be  somewhat  larger,  for  when  i0  is  small  the  curvature  of  the  relative 
path  of  the  two  stars  will  be  small,  and  such  cases  may  escape  detection. 

Applying  this  method  to  the  systems  mentioned  above,  the  results  are 
as  follows.  The  data  for  61  Cygni  are  those  of  Bergstrand*  modified  only 
by  changing  /„  from  1902  to  1857;  the  others  are  from  least-square  discus- 
sions of  the  measures  by  the  writer. 

TABLE  49. 


No. 

Star.         I     /.           p. 

• 

b 

c 

u1 

*' 

(w^+ro.)  sin'i'o 

»«.M 

J7.  )8 

41 

r  2398              1870  :   HA' 
61  Cygni        1857     106  4 
Kriiger  60      1900     149 

.5:96 
'7  77 
3  .6 

+oTooJ4 
+0.0913 
+0.0070 

o'oooQ4 
0.00048 
o  0138 

+or0} 

+O.02 
—  0.70 

+oTo584 
+o.  1665 
—  o.  192 

0.43 
0.13 
0-35 

•Nova  acta  reg.  soc.  scient.  Upsaliensis.    Serie  iv,  vol.  i.  No.  3  (1905). 


RESULTS  OF  OBSERVATION. 


IOI 


The  average  value  of  (;«i  +  w2)  sin'/'o  for  the  three  systems  is  0.30,  and 
the  most  probable  value  of  «Zi+m2  is  one-half  the  mass  of  the  Sun. 

In  spite  of  the  uncertainty  introduced  by  the  unknown  factor  sin3/0, 
there  can  be  no  doubt  that  these  systems  are  considerably  less  massive  than 
those  previously  discussed ;  for  otherwise  it  would  be  necessary  to  suppose 
that  the  average  of  three  values  of  sin'io  was  only  one-eighth ;  and  the  prob- 
ability that  sin3z0  will  be  so  small  in  a  single  case  is  only  0.13. 

The  average  mass  of  a  component  of  one  of  these  systems  is  one-fourth 
that  of  the  Sun.  Even  if  its  density  is  8  times  that  of  the  latter,  or  1 1  times 
that  of  water — a  somewhat  violent  assumption — its  surface  area  must  be 
one-tenth  that  of  the  Sun.  The  actual  brightness  of  these  stars  (using 
Kapteyn's  value  for  the  Sun's  light*)  is  as  follows: 


TABLE  50. 


61  Cygni. 

22398. 

Kruger  60. 

o  063 

O.  OO^3 

o  0031 

Companion  .  .  . 

0.033 

0.0018 

0.0009 

It  is  evident  that,  except  in  the  case  of  61  Cygni,  their  surface-brightness 
must  be  very  small,  not  exceeding  one-thirtieth  that  of  the  Sun,  and  in  some 
cases  much  less.  This  affords  further  evidence  of  their  low  temperature  in 
addition  to  that  of  their  spectral  types  (Ks  and  M) ;  and  shows  also  that, 
unless  their  surface  brightness  is  considerably  less  than  that  previously  com- 
puted for  the  average  of  stars  of  this  type,  their  density  must  be  very  great. 

13.  Distribution  of  Stars  of  Different  Spectral  Types. 

The  comparison-stars  whose  magnitudes  and  spectra  were  determined 
at  Harvard  number  over  200,  and  are  widely  distributed  over  the  sky.  They 
were  chosen  upon  inspection  of  the  plates,  principally  with  regard  to  their 
photographic  magnitude,  sometimes  also  on  account  of  favorable  position, 
but  quite  without  knowledge  of  their  spectra,  and  there  appears  to  be  no 
reason  why  they  should  not  be  regarded  as  fair  samples  of  the  stars  of  corn- 
arable  photographic  brightness. 

As  the  only  published  data  regarding  the  relative  proportions  of  the 
different  spectral  types  among  stars  equally  faint  (averaging  somewhat  below 
the  ninth  magnitude)  appear  to  be  those  derived  from  a  single  plate  of  a  rich 
region  in  the  Milky  Way.f  the  information  furnished  by  these  stars  may  be  of 
value.  The  field  in  which  they  lie  may  be  dividher  rather  sharply  into  two 
groups,!  according  to  their  distances  from  the  Milky  Way — the  mean  galactic 
latitude  of  the  centers  of  the  first  group  being  10°,  and  of  the  second  58°, 
with  only  two  individual  latitudes  over  20°  in  the  first  and  under  50°  in  the 

*With  Pickering's  value,  all  these  numbers  would  be  halved. 

fHarvard  Annals,  vol.  LVI,  No.  i,  pp.  14,  21. 

JGalactic:  Series  1-3,  5-9,  23-27,  29,  30.     Non-galactic:  Series  4,  10-21,  28.     (See  table  A.) 


102 


DETERMINATIONS   OF   STELLAR   PARALLAX. 


second.  Excluding  series  17,  1 8,  22,  31,  for  which  the  spectroscopic  observa- 
tions are  very  incomplete,  there  remain  15  galactic  and  12  non-galactic  fields 
for  discussion. 

For  comparison  with  these  there  are  added  groups  composed  (a)  of  the 
faint  stars  of  large  proper-motion  among  those  especially  observed  for  paral- 
lax (see  page  95)  and  (b)  of  those  stars  among  the  latter  whose  observed 
parallaxes  exceed  oTio  (after  allowance  for  the  parallax  of  the  comparison- 
stars).  We  thus  obtain  four  groups,  comparable  in  apparent  brightness 
(especially  from  the  photographic  standpoint),  but  differing  in  other  respects. 
The  first  three  are  mutually  exclusive;  the  fourth  a  sub-group  of  the  third. 

The  numbers  and  percentages  of  stars  of  the  different  spectral  types 
in  those  groups  are  as  follows.  As  in  the  recent  Harvard  work,  GS  is  counted 
vvith  K,  KS  with  M,  etc.  The  photographic  magnitudes  have  been  derived 
from  the  visual  by  means  of  King's  table  of  corrections  for  the  different 
spectral  types.* 

In  the  case  of  pairs  of  stars  with  common  proper-motion,  only  the 
brighter  component  is  counted ;  and  in  forming  the  percentages  for  the  first 
two  groups  the  few  stars  not  observed  spectroscopically  are  ignored. 

TABLE  51. 


Group. 

Mean  magnitude.       Number  of  stars. 

Not 
ob- 
served. 

Total. 

Percentage. 

Visual. 

Photo- 
graphic. 

A 

F 

G 

K 

M 

A 

F 

G 

K 

M 

Galactic  

9  38 
8.9) 
7.89 

8.21 

9.84 
9.70 

«95 
9  J8 

}4 
5 
o 
o 

35 
10 
o 

0 

16 
36 

10 

I 

43 

'9 
7 

3 

o 

8 
8 

4 
t 

131 

88 

J5 
M 

1 

o 
o 

21 
12 
0 
O 

M 

43 
40 

9 

36 
35 
28 
18 

0 

4 
33 
73 

Non-galactic  
Large  proper-motion  . 
Large  parallax    .... 

The  steady  increase  of  the  percentage  of  red  stars,  at  the  expense  of 
the  white,  is  very  striking. 

As  regards  the  last  two  lines,  the  explanation  clearly  lies  in  the  fact 
brought  out  in  §  10  (pages  96,  97) — that,  among  stars  of  moderate  or  small 
intrinsic  brightness,  faintness  and  redness  go  together. 

The  stars  of  the  last  group — invisible  to  the  naked-eye,  and  with  very 
large  parallaxes — are  of  necessity  intrinsically  very  faint,  and  therefore 
almost  all  red. 

For  those  of  the  third  group  this  requirement  is  less  rigorous,  and  so 
more  yellow  stars  (Type  G)  are  included. 

The  region  of  space  within  which  such  stars  must  lie,  in  order  to  appear 
within  given  limits  of  visual  magnitude,  is  several  hundred  times  greater  in 
volume  than  the  corresponding  region  for  the  fainter  stars  of  Type  M ;  and 
it  might  seem  that  they  ought  therefore  to  be  correspondingly  numerous. 
But  the  conditions  of  inclusion  in  group  3,  which  require  also  a  large  proper- 
harvard  Annals,  vol.  ux.  No.  5,  p.  152. 


RESULTS  OF   OBSERVATION.  103 

motion,  demand  for  these  stars  an  actual  cross- velocity  so  great  that  only 
a  small  percentage  of  the  stars  within  the  aforesaid  region  exceed  it;  and 
this  is  probably  the  reason  why  the  numbers  of  proper-motion  stars  of  Types 
G,  K,  and  M  are  approximately  equal. 

This  may  also  explain  the  absence  of  faint  stars  of  large  proper-motion 
and  spectra  A  or  F.  Stars  of  these  spectral  types  are  presumably  brighter 
intrinsically  than  those  of  Type  G,  and,  to  appear  of  the  required  brightness, 
must  lie  at  such  great  distances  that  even  the  highest  velocities  occurring 
among  the  stars  do  not  change  their  direction  from  us  by  so  much  as  i"  a  year. 

If  the  limits  of  the  group  were  widened  so  as  to  admit  stars  of  smaller 
proper-motion— or  brighter  stars  of  the  same  proper-motion — we  might 
expect  the  percentage  of  stars  of  Type  G  to  increase,  and  stars  of  Type  F  to 
appear,  followed  by  those  of  Type  A. 

The  data  for  the  comparison-stars  show  the  usual  excess  of  stars  of 
Type  A  in  the  Milky  Way  as  compared  with  the  regions  outside  it;  but  the 
actual  percentage  of  such  stars  is  surprisingly  small.  The  ratio  of  the  number 
of  stars  of  this  type  to  that  of  Types  F,  G,  and  K  together,  is  in  the  present 
case  0.41  for  the  galactic  fields,  and  only  0.07  for  the  non-galactic,  whereas 
the  similar  ratios  for  stars  of  all  magnitudes  down  to  8.2,  as  determined  from 
counts  made  on  a  great  number  of  plates  at  Harvard,*  is  2.10  in  the  Milky 
Way  and  0.70  outside  it. 

For  the  single  plate  of  a  rich  galactic  region  on  which  fainter  stars 
were  investigated,  the  corresponding  ratio  for  stars  between  the  estimated 
magnitudes  8.5  and  9.5  (i.  e.,  those  shown  with  a  longer  but  not  with  a 
shorter  exposure)  is  2.9.  It  is  not  certain,  however,  that  this  one  region  is 
typical  of  the  whole  Milky  Way.  It  is  hard  to  explain  so  great  a  discrepancy. 
There  seems  to  be  no  reason  whatever  why  there  should  have  been  any 
discrimination  against  stars  of  Type  A  in  picking  out  comparison-stars  in 
fields  where  all  the  spectra  were  wholly  unknown.  The  fact  that  the  selection 
was  made  on  photographs  would  favor  the  whiter  stars  at  the  expense  of  the 
others ;  but  the  same  is  true  of  the  Harvard  plates  from  which  the  data  just 
referred  to  were  obtained;  and  the  fact  that  the  estimates  of  spectral  type 
were  made  in  both  cases  at  the  same  observatory,  and  on  the  same  system, 
makes  it  very  improbable  that  large  systematic  differences  in  classification 
exist.  It  is  very  desirable  that  the  question  should  be  settled  by  determina- 
tion of  the  spectra  of  a  much  greater  number  of  faint  stars,  well  distributed 
over  the  sky. 

It  is  possible  that  a  diminution  of  the  relative  proportion  of  stars  of 
Type  A  sets  in  at  about  the  ninth  magnitude,  similar  to  that  found  for  Type 
B,  in  passing  from  the  brighter  to  the  fainter  naked-eye  stars;*  but  it  would 
be  premature  to  suggest  any  explanation  until  the  reality  of  the  phenomenon 
is  better  assured. 

'Harvard  Annals,  vol.  IMI,  No.  i,  p.  21. 


104 


DETERMINATIONS  OF   STELLAR  PARALLAX. 


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DETERMINATIONS  OF  STELLAR  PARALLAX. 


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DETAILS   OF  THE   OBSERVATIONS. 


107 


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TABLE  C,  SERIES  XIa.     STAR  13;  Lai.  21185.     y  Co-ordinates. 


Stars. 

Standard 
mean  of 
191  and  194 

191 

194 

258 

260 

268 

397 

405 

426 

2 

6 
9 

19.74821 
11.78118 
31.59270 

63'75 
68525 
53881 

86468 
87711 
64660 

29722 
34156 
20162 

65062     70200 
70130     73778 
56754     58585 

71560 
77020 
62460 

64839 
70158 
54908 

66138 
67678 
46458 

A        20.37582 

28496 

46669 

93828    29759 

33351 

34162 

27082 

23610 

Average  residual  . 

51 

3' 

45 

45 

oo 

28 

31 

25 

Residuals  for  A...      +19       -19     -807     -817     -845 

1                                                 : 

-2835 

-2902     -3433 

Reduction  constants      y/,  =  +o  .  33054^2  +o  .  382993*8  +°  •  2864751,, 

Plate. 

Observed. 

P.  M.  to 
1904.0 

Corrected 

Equations  of  conditions.       \    o—cl 

o-c, 

191 

+oTo33 

-oT289 

-oT256 

x—  o.o6iy—  o.294T=—  0^256      +0^040 

+oTo4i 

-0.033 

—0.242       —0.275       x—  0.051?—  0.2561=—  0.275       +0.009 

+O.O1O 

258 

—  1.419       +1-379       —0.040      *+o.29iy+o.595T=—  0.040       —0.041 

—0.041 

260 

—  1.436       +1.417       —0.019      *+o.299;y-i-o.  585*=—  0.019       —0.018 

—0.018 

268 

-1.486       +1.493       +0.007 

3c+o.3i5y+o.57iT  =  +o.oo7       +0.013 

+0.013 

397 

-4.984        +4-735        -0-249       x+o.999J-o.078r=  -0.249       -0.024 

—0.025 

405 

—  5.102       +4.863       —0.239 

*+i  ,O26>'+O.O27T=—  0.239    i  —0.049 

—0.050 

426 

—6.035       +6.  105 

+0.070 

*+  1.  28831+0.  596r=  +0.070      +0.070 

+0.068 

1 

Assumed  P   M 

—  A    7^O 

.  .  (DVV)        11712 

1  1704 

!                   i                                                                         i 

Normal  equations. 


Solution  I.        Weight.       Solution  II.        Weight. 

+8.OOOX+4.  io6y+i.7497r=  —  iTooi    x  =— oTioS^oTo^  3.61  x  ——  oTi99±oToi2    6.01 

+4.106  +3.989  +1.277  =-0.389    y  =-0.002=0.025  1.76  

+  1.749  +1.277  +1.540  =+0.169    T  =+0.335*0.031  1.08  ir  =+0.335*0.028     1. 10 

r0=-  =4=0.033  rc=  ±0.030 

Measurer's  notes — 260,  Plate  spattered  with  fine  drops. 

All  quantities  in  this  table  are  in  reseau  intervals  of  175^8  unless  otherwise  stated. 


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Solution  II. 
A  B  Weig 
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+0 


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=0.055  ±0.063 


. 
A  B  Weigh 
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y  =+0.110*0.046  +0.089=0.059  1.14 
T  =+0.072*0.027  +0.054=0.034  3.34 
r0=  =0.049  =0.062 


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-  .a  r 

r(N  3     -     .  " 
Oj<  ^M^ 

H 
1 

^"    •<*•        CM  05         IO    rA       «*    M          UN 
"*•       CM  r™               ^"       CO   M         f^\ 

rr\ 

M  ON 

" 

HTf  a\       T(  M       \£)  ON        ON 
O  -       00  M       VO  UN        m 
rr  u-\       OO   r<N        ir\\o        ^p 

m-         O-*        (S-        OOuNxO 

r^      S 

u 

1 

0 

M  m     co^-      eo  -      COCJ       « 
«-                     ff\ 

-SS'S 

"'"oo' 

" 

ON      6 

vo  ~  a  - 

r**  ^  ™   i  /v 

Z°^i 

T 

0 

r-o>      1  1»     r-w       oco      UN 

T-CM             M     -                   CO             UNT-             M 

•H-l 

T 

0 

M  WV  M  W 

0^88 

is 

«N 

K 

O  00         ON  M         O   ••          UN  ••         M 
Mrri        *-O        ,2^"       OC'UN        CP* 

l  ++  l 

+  1 

S 

O  O  9  O  O  O 

1  +++  1    1 

w     a 

S"fi+ 

0*    ^      '^"    fcfl     yj 

Sgw,;0 

u 

1 

0 

-ON    ^-r»     vo'o     oo     10 

—   M        OJr—               irxiO—        ^| 

^R2; 

.  #    VO  Vp  00 

1  1  + 

Equations  of  con- 
dition. 

II    II    II    II    II    II 

<3\  O   ON  t^«  r*"\  UN 
•—  i   M    UN  mCQ   ™ 

1 

« 

00  ^O         ONOO         ff\  M         eft  O1        ^ 
ON  r*«      m  ^      M  ^      ••  op       ON 
O\  rn        u\  -tj-      \O  O         roiVO         ^ 
MO        C2.rrx        ov^         o—         "^ 

O  O  O  O  O  O 

—  co  oo  ^  o  ^r 
M  —  L/N  —  •  ir\  Is-. 

—   ^-   Tf  -^-CO  CO 

r-     ^        ' 

•g            *Jjt- 

O  us      t>.  M       "•  ^       ON  M       ON 

—   M          t^.^        t^.—         MUN        MUN 
—    t-»         ONt^*         OUN         Mt>»         MO 
M  VO         t-^OO         I**.  O         -^  m        °V-L 

Average  residual  
a  
b  
c  

O  0  0  0 

1  1  1  1  ++ 

M  H  «  «  H  « 

§ 

~t->                                              QJ 

CO                         +^  vo 
P*  cfl  r^ 

-ti  ^i  d  ^  ^ 

•a 
a 

CJ 

HisH 

1 

o 

-  o      -  i-      r^o      c 

MM          M    m         M    •"          ""    "•          M 

0  0  0  0  O  O 

-d 

S 

i 

« 

d 
W 

oo   wS   wo   ww  £ 

3o 

mi 

0  0 

-  —  ff\ 

UN 

co 

d 

H 

c\^     oloT    odod     "So7    v? 

o  o  o  o  o  o 

++++  1  1 

Bonn  Durch- 
musterung. 

M 

M 

S 

ON  t^.       ONGO       OO  00        ONX        !>• 

1 

Plate,  i  Observed. 

Tf  ^J"  N    O^  ^  W* 
•«•  ^VO  CO   t^  O\ 
O  O   UN  UN  u\  UN 

1 

•o 

a 

»i 

< 

0 

O  m       TTVO        m\O        M  -        r- 
C^  ON       ON  O^       O  Is*       ON  t""»       O* 
r^  r*«       t^*  t^-     co  r^      t*^  t**      r^« 

O  O   O  O  -  - 

1  +++++ 

o 
MM          MM           M—           M—          M<< 

+                                              -o 

•3-VO   •«•  -  VO_  - 

• 

E 

tn 

—  rr*       if  UN       r>.oo        M  \O       ^J 

134 


DETERMINATIONS   OF   STELLAR   PARALLAX. 


laJ 

Is! 


o 

I 


•6 
I 


^T 


P 


««    .t 

m    .  B  * 


O 

r» 

M 

*O 


I 


I 


^O        O)    O          M  IO    ^       O>    O\         O> 
'fCJM  O*—    M         COM  *^ 

O       O   O         O  O   O        O   O         M 


5-    -s    S 

So      o  o     o 


oo  UN    Nor*.      o\- 

*+  v-CSICN         —   ^- 


\  «  -^J-OO  M    'TOO  -^"O  M 

-uN  «*r»  —    IN.  «*N  t^-O  rf\ 

•  ^-  r>.  ^  "^vo  —  if\  «  « 

c\  oo  oo  "<r  *^QQ  *«r  o\  vb 

O  **N  ^-  er\nO\  00-  ON 


o  •» 

*!>.« 

"  SS 
+  1  + 


O\  OS  00  *O  _O\  O\  M  00  VO  00 

—  —  r^-  1*\  oo  o  oo  **^o  f*^ 

»-    Q  O\  i^  00   **•*  ^*  ^"  ^  ^ 

oo—  oo  ^or>-o  t/-\o  oo 

r-.o  ^  ^  «  —  o\  r^.—  oo 


U 

0 


f 


r« 
tt 
O 


vo      g 


e,  «      o 


00 
*f\ 


00  ^^        <S   ^00         O  00 
-- 


. 

—  rr\ 
rt  ir\  -^ 


"R 
1  + 


S.& 

wis 
-* 

+  1  i 


««*•    CMW      n —*>     o>  m     >^ 


g.R    ^2 

f?  SK  vS1^  <§,?;    s 

«•»  «•          r^f       CM »-  *O          CT\CM        CO 


0?    1^ 

0000 


+  1  + 


pa^-SS 
«« 


+  i  + 


^C—          OO          Ci^I^        rr\  ~t         -T 

o?       &  onco       i^J     ^O( 

O«         Sr^       vpr»5        rA—         ^-( 


* 


M 


it    i 


oo  co 
-r  »r 

00r>- 


-  a>  o       o 


^<      ^ 


ONOO       00  O1       O>00  ON      00  O*       ON 


00   ON      00 
VOVO        \O 


w- 

o\     <« 
• 


dual 


Ave 


bo    • 

•3    t^M 


„  u    I-.  ni\^ 

§  5853 
s  *d  d  d  d 
•3  HI  II  HI  HI 

r/3          -fl-MOO 
fA  rf\  ""* 

o  q  « 
*d  d  d 


lo 


S' 


I 

.s 

w 

o 


II   II   II   II 
«  ^  h  w 


°^~^§ 
*d  d  - 


a,  - 


1 


s 


»      l> 

O        k 
O         ~, 


eq 


No 


bn 

c 


•a 

3 


u 

C 


a 

» 


£ 
.S 

tf   « 


3  «  r 

d  .5 

-8  j»  o 

H  •  • 


O\  r»   5\ 

o  r^r~ 

d  «  o 


8"?, 


^     <£•  ">*?> 

T  r*i 

I    l-s« 

§      §  .§2 

M  •»     .t!.§ 

s    fe  II 

Jr  H      9.P 

1? !  !a 

St*  ^'S 


rr\o»r>—   rr\M«-rt 
OOOOOOOO 

*d  d  c id  d  d  d  d 
I    I  ++  I    I+  + 


00  ^O  *O    —  OO   t^  -^  r 


ddddd 

1    1  ++++  1    1 


II   II   II   II   II   II   II   II 
fc  h  It  b.fc  fc  fcfc 

w\o    —    M\O    wl^t^ 


n 
r^ 


--  00  --  -  - 
1  1  1  1  ++++ 
HHHKHKHK 


O\  ITMO  »O   O 
00  >O   mCO   « 


--OOOO-- 
1    1    1    1  ++  +  + 


8 

T 


OOOO""   —  Mfl 

I  + 1  I  I  I  I  I 


CO 

d 

I 

$ 

•a 


Assumed 


DETAILS   OF  THE    OBSERVATIONS. 


135 


TABLE  C,  SERIES  XXX.  STAR  42;  Lalande  45755;  23hi6°?8,  +43  "32';  P.  M.  +o?os8,  +oT22,  =oT68. 

Approximate 
solution. 

1= 

o  o       o  o     o  o       o  o      o 
*d 

'3 

HH          ^         O  M3 

I  5    I? 

•43      o      oo 

3         "n          "fl    71 

^co-          «     .1      ? 

1^*0  o                     o       o 

*d  d  d 

g      II   11   II 

3           ... 

Q*          *•    O    ffk 

•u       |  X^. 

g     [f^vc 

g     °S% 

^     ?++ 

8  N  -                 **  SN^  N 

O   rr\  UN                    -Q 
O   O   O\                    (W  UN  w   rr\ 

+TT     ^^ 

HH       —  m  M  rA 

a     ^o  o  o  o 
.2     "o  d  o  d 
3      -H  •«  •«  -H 

•2        OVO  1^ 
O        i^  T(-  m 
W        OOO 

'odd 

II  II  II  II 

«  ?\  N  w 

Observer's  notes  —  342,  Sky  thick;  347,  Clouds  at  end. 
Measurer's  notes  —  342,  Fogged;  347,  Exp.  3  faint;  4  invisible;  349,  Images  diffuse. 
The  proper  motions  obtained  by  rejecting  Star  i  are  given  under  the  heading  n'. 
All  quantities  in  this  table  are  in  reseau  intervals  of  175^8  unless  otherwise  stated.  Numbers  in  bold-face  type  are  negative. 

> 

O  ^"        OO   !*>•       CO  00        CO   UN        ON 
Q  C\|       co  ~       mm      h*O\       r* 

koo      coo     ^-  —     oo      o 

*0 

a. 

d 

flffj 

1 

COO        OOUN       o   rr\       F—00        00 
N        C300        ^TO        OI^        O 

"1 

H 

11  11  If  II  1 

^     B 
r-  .  B   ~ 

%ST+w 
-ww 

u 

1 

a 

—  VO        ON        r-OO        h*O         t^- 
N    N        O   UN       O-CO        «* 

^~ 

ON  r* 
r»  n 

H 

II    1    II    II   I 

1 

"a 

T 

O 

*S    5?    So    KJ    | 

-00 
—  —  00   Ol 

vo  ooo  r^ 
+  1  + 

* 

N    rr\       00   !*>•        —   N          ^"O        OO 

nj  _L  CX  ^ 

T 

« 

rTN  UN 

O  ON 

EK 

+++ 

ff\  O             UNQ             OUNOOrTN            ^ 

K               fYNrrNOO^-        NUN       rr\I>.       O 
UNON       OOuN        OO         O\«N        UN 
O  t"~*       ^*  O         t^"  ••         t**  ^       •• 

| 

T 

0 

«   rA       Jg   ^       CJCM        h*   0         ^ 

o  s?\ 
n  O 

^  C^OO   ^f 
—   M   — 

+    1    + 

ft 

T 

^J"  ^  rr\  CTi  P*.  O    "N 

j^o  £-3-3  fO 

o  o  o  o  o  o  o 

d  d  d  d  d  d  d 
1  ++  1    1  ++ 

VO 
CTl 

H 

OON        OOO          ON         —   —         •^> 
N  CO          ^S5         "^    '***         *^"00           O 

10 

1 

0 

<To  o  c?  o  o  o 

1 

S>  "a 
^~?) 

T 

UNO     toco     r-^  UN     r-eo      t^. 

o"S 
•  #    a\  —  oo 

M    0        >•£>   0 
rri        "™  5 

+   1   + 

o  o  o  o  o  o  o 

1  ++  1  1  +  1 

« 

o1  o  —  o  oo  o 

H 

O    rr\        O  CO          N    ••          ^s  UN        [*>. 
CO   UN        N   t^.        -<*-OO         UN  -        CO 

0  0  0  0  0  0  O 

OG         c 
~   O     -     . 

|||  || 

V) 

1 

0 

*t  co       WN  UN      r-co       r-  ^j-     r^. 

O     UN 

*    o\  -co 

ri*  *l 

i  +7 

Equations  of 
condition. 

H  H  II  II  H  n  II 

fc  fc  fe   k  N  N   N 
m  ?  i^oo1  >^  0*0? 

CO  00    ^  •**••«•  ONOO 

x-s 

• 

^  O\        ••  O         ~™   Q        CO   N          UN 

O  0  0  0  0  0  0 

^  H-++  1  1 

-  c5?VC  J?'  -    O\  5 
r*    —  GO    O    —    M    l^ 

V£>\O   C    -    -    -"T  •* 

w|l^ 

r^-co       UN  -       j-  N      co  co       N 

Average  residual  
a  
b  
c  

o  o  o  o  o  o  o 

Corrected. 

—   ON  &*  M   f^  rAOO 
VO   —    -f  <s    -<J-^O    — 

d  d  o  d  d  d  d 

1 

d 

1 
w               1«  "^ 

S  " 

cor-.      ONN       •^•^f      NO       ON  t 

—          (NN           NN           —    N           —     l 

1 

a 
SB 

bo 

rt..      UN                                                                                      fV.                            QQ 

faO     <O     fa<     fafo     fa 

o    . 

•H    „, 

cTioo  ^^o  i^  r^  o\ 

r^  r*>  o  O  O  N  N 

o 
d 

OO         rftrrt        NOO         Nt-*VO 

o—      uNr«-      t->.aN      N—       UN 
ON  o       ON  Os      ONOO       O  O       r*» 

o  o  o  o  o  o  o 

++  1  1  1  1  1 

Bonn  Durch- 
musterung. 

I 

O1  ON      ON  ON      ONOO       Os  O*t      r^* 

Observed. 

w.  rc\  O  GO  —  r^  — 
O   O^O   Tf  »r\  1^  h* 

Assumed  P.M.  andAx 

d 

+  *   **   ^*      *   *-s 

o  o  o  o  o  o  o 

1  ++++++ 

o 

J2 

SSaStaS* 

d 

M 
•M 

w 

r*  m       —  TT       i^QO       SO  r^      <J 
h 

136 


DETERMINATIONS  OF   STELLAR   PARALLAX. 


M 
T 


DETAILS   OF   THE    OBSERVATIONS. 


137 


TABLE  C,  SERIES  XXXII.     STAR  45;  ij  Geminorum;  6"8™8,  +22°32';  P.  M.-o?oo5,+o?o2, 

Observed  with  Color-screen. 


=  0.07 


Plate  233  (Stand.) 

Bonn  Durch- 

Txp^V"8           Plate  236          Plate  380          Plate  381 

Plate  383 

Stars. 

musterung. 

Hour-angle  -4°      '9°4,  Mar.  10    1904,  Oct.  24     1904,  Oct.  24 
Obs   R               ExP-  4.  5m           Exp.  4,  5™          Exp.  3,  5" 

1904,  Nov.  7 
Exp.  4,  5i» 

Hr.-ang.  +  5m    Hr.-ang.  —  lo01;  Hr.-ang.  +  15""    Hr.-ang.+3m 

Obs.  R               Obs.  H 

Obs.  H              Obs.  H 

No. 

Mag. 

' 

f 

r 

+22°  1237 

9.  i 

18.183 

18.73825 

74989 

86715 

88542 

86034 

2 

+  22       1247 

9-5  :  25.736 

23.91850 

94262                  09259 

07573 

04952 

3 

+22       I25O 

8.8     13.410 

26.38734  |         38764 

48780 

52940                 50501 

A 

+22       1241 

3-2 

20.271 

21.04475 

05932 

18586 

19510 

16949 

Average  residual 

40* 

26* 

37 

25 

35 

Residi 

ials  for  A  .  . 

0 

-24 

-29 

+30 

+  9 

Reduction  Constants  ^  =  +0.5929  x, +0.3270  £2+o.o8oi  x~. 


Plate. 

Observed. 

P.  M.  to 
1904.5. 

Equations  of 
condition. 

o-cl 

o-c. 

Weight. 

233 

236 
380 
381 
383 

oTooo 
—0.042 
—0.051 
+0.053 
+0.016 

—  OTO22 
—  O.O22 
+O.O22 
+O.022 
+0.024 

x-o.  3\jy-o.  9631= 
x—  0.31  \y—  o.9727r  = 
ar+o.3i4ji+o.86&7r  = 
*+o.  314^+0.  866V  = 
*+o.353;y+o.7257r  = 

—  OTO22 
—  0.064 
—  O.O29 
+0.075 
+0.040 

+oTo2  1 

—  O.O2I 
—  0.050 
+0.054 
+0.024 

—  oToi4 
—0.056 

—  0.021 

+0.083 
+0.048 

i 

i 
i 

i 

Assumed  P.  M  .  .  . 

-69 

(ovv) 

6874 

9521 

Epoch  mean  equations. 


Solution  I. 


x— o.3i4y— 0.967^=  — 0^043 
£+0.339  +0.810  =+0.019 


Solution  II. 
A-  =-oToo8 


T  =+0.035  —  0.3631  ±oToi7 
rn  =  ±0.032  r0  =  ±o!o33 

With  Boss's  P.  M.  y=  +0^005 
ir  = +0.034 

Observer's  notes— 233,  Control  slow;  380,  381,  382,  seeing  poor;  382,  High  wind. 
Measurer's  notes — 233,  Images  elliptical;  380,  381,  Diffuse;  381,  Image  4  of  A  very  discordant 
exp.  4  therefore  rejected. 

All  quantities  in  this  table  are  in  reseau  intervals  of  175^8  unless  otherwise  stated. 


138 


DETERMINATIONS   OF   STELLAR   PARALLAX. 


Ilip 


2,1 


I  « 


00       « 


33 


ON,  O  —  ^  M  "''OO  00  M 

t^»  —   **"*  \O    O\  ^PO  O  00 

^vot^  Mt^.  \O^-  ^t  M 

Or^O  ^    Q  ~S?  MO> 

ON  ^-  m  O    ON  —    O  ft   ~ 


»^00          r*    rr\ 

5,     2S- 


M^D 

•-  ON 


1  ++ 


1  + 


^  2>    ?J2S    SS1    P>r?     «S?        S  ? 

VO    »rv        O  00   O          " 


f^r»        O  rt 
m-         r*   © 

—  ^         -^ 


i  ++ 


u 

*r  — 
O  ffN 


O  ^ 
trs  •• 
O\  *r\ 


1    1 


eg 


N         ^00         O 


r^-         " 
ON        - 


•s-s     has   -"s 

M  9-;         *^C     ,!T?I 


j!t 


P-  C4   rrt 

S" 


» 


i  fa 


XO 


i   *         000 
.   —          w\  « 


>^  m       t*~  t- 
1^-00       <«  t- 


I*M^       or^O- 


•*T-      —  «A      o  o 
-        -  -        n  n 

MA       n  "-.         l>i 


O\  t^.      00  00 


"3 

3 

•6 


n— 

f»*,  rr\ 

Jf 


1 

ii 
a 

e 

ri 

II 

ts-sHf 

i 

c; 
o 

ffl 

O 

o  o  o  o  o  o 

1+  1  1  ++ 

m 

J3 

-  8  -  S  £"£" 

0  0  0  0  O  O 

1  +  1  I++ 

Star  A. 

T 

si  ?  If  if 

d  d  d  d  o  o 
1  1  +  1  1  + 

00  00  O  t"*  O\  "*» 

U"\  TT  ™"  O  WN  !**• 

000  

r 

0  O  0  O  O  O 

1  1  1  +++ 

"5 

d 

Ml  Ml  II  II 

S-&-  t>&0^ 
rr\  ^J-  O  VO  05  fl 
OO  00  Q\  <^CO  00 

O. 

d 

3 

a 

conditio 

o  o  o  o  o  o 

1  1  1  +++ 

<&  Q  —  *r\  rr\  rr\ 

W 

O  0  O  0  0  0 
1  1  1  +++ 
H  H  H  «  «  H 

0 

US 

u%  IA  tr\  ir\\S  O 
O  O  O  O  O  O 

p. 

S 

1 

O  O  O  O  O  O 
1  1  1  +++ 

o 
1 

5-f^f?^ 

1 

o  o  o  o  o  o 

1  +  1  I++ 

* 

o 

181?!= 

S 

.1 

"b  6  o"  d  d  d 

^ 

•o 
V 

l< 
V 

i 
5 

i 

4 

*r\00  »^  «  ^  lfN 
rr\  m  ^00  00  00 
M  M  M  rfN  m,  <r\ 

1 

s 

I 

.n 

O 


DETAILS   OF   THE  OBSERVATIONS. 


139 


TABLE  C,  SERIES  XXXIV.     STAR  48;  y  Serpentis;  I5h5i™8, 

P.  M. +0*021,  —  i?3O,  =  1^33.     Observed  with  Color-screen. 


Plate  239  (Stand.) 

Plate  252            Plate  302            Plate  306 

Stars. 

BonnDurch-           ^pM^° 
tnusterung.         Hour-angle  +10" 
Obs.  R 

1904,  Mar.  31 
Exp.  4,  5ra 
Hr.-ang.+25m 
Obs.  R 

1904,  June  21 
Exp.  4,  5m 
Hr.-ang.+24m 
Obs.  R 

1904,  June  25 
Exp.  4,  5° 
Hr.-ang.  +9° 
Obs.  R 

No. 

Mag. 

1 

« 

X 

0  —  C 

X 

o—c 

X 

O  —  C 

i 

+  16°  2841  ;    9.5 

21  .580    :       8.O9IOI 

17504 

83    '    °935°      63    ;     12360 

Kg 

2 

1  6    2846 

95 

23.293          16.89078 

98265      17    j    89900      32 

93244 

93 

I 

1  6    2847 
16    2851 

9.2 
8-5 

24.828 
28  .  04O 

18.37451 
26.62146 

47405 
7357° 

20 
86 

38344 
63739 

62 

31 

42395 
68582 

77 

5 

16    2850 

95 

20.972 

22.93208 

01425 

29 

9454<> 

21 

97025 

36 

8 

16    2852 

94 

22.OO3 

31.45126 

539"  5       36        47'8i 

49 

49758 

9 

4 

15    2931 

93 

14.077 

18.43422 

48358         2         44554       55         44550 

65 

7 

15    2936 

9.0 

15.656        29.76559 

82278       66 

78418 

85        78749 

91 

A 

16    2849 

3.8 

2O.315 

20.95984 

03841 

5 

97296 

107        99519 

103 

Average  residual  

il* 

31* 

fio* 

68* 

a  '             o.oo 

+     10.13             +  78  13              +     57.83 

b  o  no 

+  464  89             —   n  01             +  356.47 

c  

O. 

—  1795.                 —209.                 —5021. 

Plate. 

Observed. 

P.  M.  to 
1904.5. 

Equations  of 
condition. 

tt 

o—c, 

0  —  C2 

239 

252 
302 
306 

oTooo 
—0.009 
+0.188 
+0.181 

+0^090 
+0.073 
+0.008 
+0.005 

x—  0.3  13^+0.9  1  2?r  = 
*—  0.25231+0.  738*-  = 
*—  0.029^—  0.495*-  = 
x-o.oi7y-o.553T  = 

+oTogo 
+0.064 
+o.  196 
+o.  186 

+OTO2I 
—  O.O2I 

+0.007 
-0.008 

—  oTo44 
—0.070 
+0.062 
+0.052 

Assumed  P.  M.  .  . 

+0.288 

(pvv) 

995 

13384 

Epoch  mean  equations. 
x— 0.282^+0. 8251-  =  +0^077 
*— 0.023  —0.524   =+0.191 


Solution  I.  Solution  II. 

x  =+ori47  x  =+0:134 

r   =—0.085+0.  I9}'=>=o"oi  I  

r0  =  ±0.015  r,,=  =oTo45 

With  Boss's  P.  M.y  =  +oToi5 
»=— o  081 

Observer's  notes — 302,  Sky  pretty  thick. 

All  quantities  in  this  table  are  reseau  intervals  of  175^8  unlessotherwise  stated      Numbers 
in  bold-face  type  arc  negative 


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